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Distance 2-restricted optimal pebbling in cycles

  • Let G be a graph and let δ be a distribution of pebbles on G. A pebbling move on the graph G consists of removing two pebbles from one vertex and then placing one pebble at an adjacent vertex. Given a positive integer d, if we can move pebbles to any target vertex v in G only from the vertices in the set Nd[v]={uV(G):d(u,v)d} by pebbling moves, where d(u,v) is the distance between u and v, then such a graph pebbling played on G is said to be distance d-restricted. For each target vertex vV(G), we use m(δ,d,v) to denote the maximum number of pebbles that can be moved to v only from the vertices in the set Nd[v]. If m(δ,d,v)t for each vV(G), then we say that δ is (d,t)-solvable. The optimal (d,t)-pebbling number of G, denoted by π(d,t)(G), is the minimum number of pebbles needed so that there is a (d,t)-solvable distribution of G. In this article, we study distance 2-restricted pebbling in cycles and show that for any n-cycle Cn with n6, π(2,t)(Cn)=π(2,t10)(Cn)+4n for t13. It follows that if n6, then π(2,10k+r)(Cn)=π(2,r)(Cn)+4kn for k1 and 3r12. Consequently, for n6, the problem of determining the exact value of π(2,t)(Cn) for all t1 can be reduced to the problem of determining the exact value of π(2,r)(Cn) for r[1,12]. We also consider Cn with 3n5. When n=3, we have π(2,t)(C3)=π(1,t)(C3), since the diameter of C3 is one. The exact value of π(1,t)(C3) is known. When n=4,5, we determine the exact value of π(2,t)(Cn) for t1.

    Citation: Chin-Lin Shiue, Tzu-Hsien Kwong. Distance 2-restricted optimal pebbling in cycles[J]. AIMS Mathematics, 2025, 10(2): 4355-4373. doi: 10.3934/math.2025201

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  • Let G be a graph and let δ be a distribution of pebbles on G. A pebbling move on the graph G consists of removing two pebbles from one vertex and then placing one pebble at an adjacent vertex. Given a positive integer d, if we can move pebbles to any target vertex v in G only from the vertices in the set Nd[v]={uV(G):d(u,v)d} by pebbling moves, where d(u,v) is the distance between u and v, then such a graph pebbling played on G is said to be distance d-restricted. For each target vertex vV(G), we use m(δ,d,v) to denote the maximum number of pebbles that can be moved to v only from the vertices in the set Nd[v]. If m(δ,d,v)t for each vV(G), then we say that δ is (d,t)-solvable. The optimal (d,t)-pebbling number of G, denoted by π(d,t)(G), is the minimum number of pebbles needed so that there is a (d,t)-solvable distribution of G. In this article, we study distance 2-restricted pebbling in cycles and show that for any n-cycle Cn with n6, π(2,t)(Cn)=π(2,t10)(Cn)+4n for t13. It follows that if n6, then π(2,10k+r)(Cn)=π(2,r)(Cn)+4kn for k1 and 3r12. Consequently, for n6, the problem of determining the exact value of π(2,t)(Cn) for all t1 can be reduced to the problem of determining the exact value of π(2,r)(Cn) for r[1,12]. We also consider Cn with 3n5. When n=3, we have π(2,t)(C3)=π(1,t)(C3), since the diameter of C3 is one. The exact value of π(1,t)(C3) is known. When n=4,5, we determine the exact value of π(2,t)(Cn) for t1.



    Let G be a graph. A distribution δ of G is a mapping from V(G) to the set of nonnegative integers. For each vertex vV(G), δ(v) denotes the number of pebbles distributed to v. A pebbling move consists of removing two pebbles from one vertex and then placing one pebble at an adjacent vertex. Let δ be a distribution of G, and let H be an induced subgraph of G. For convenience, we use |δ(H)| to denote the number of pebbles distributed onto the vertices of H. For each vertex vV(H), we use m(δ,H,v) to denote the maximum number of pebbles that can be moved to v from the vertices of H by using pebbling moves.

    Given a positive integer d and a distribution δ of G, if we can move pebbles to any target vertex v in G only from the vertices in the set Nd[v]={uV(G):d(u,v)d} by pebbling moves, where d(u,v) is the distance between u and v, then such a graph pebbling is said to be distance d-restricted. We use m(δ,d,v) to denote the maximum number of pebbles that can be moved to v only from the vertices in the set Nd[v]. That is, m(δ,d,v)=m(δ,H,v), where H is a subgraph of G induced by Nd[v]. A distribution δ of a graph G is (d,t)-solvable if m(δ,d,v)t for each vertex v of G. The optimal (d,t)-pebbling number of G, π(d,t)(G), is the minimum number of pebbles needed so that there is a (d,t)-solvable distribution of G. A distribution δ of G is said to be optimal (d,t)-solvable if it is (d,t)-solvable and |δ(G)|=π(d,t)(G). When d is not less than the diameter of G, the subgraph induced by Nd[v] is G itself. In this situation, π(d,1)(G)=π(G), which is known as the optimal pebbling number of G, see [2,5,7,8,9,11] for references. Moreover, π(d,t)(G)=πt(G), which is the optimal t-pebbling number, see [12,14] for references.

    In 2018, the distance-restricted graph pebbling was first proposed by Chen and Shiue [4], and they showed that π(2,2)(C3)=4 and π(2,2)(Cn)=n for all n4. In 2020, Shiue [13] studied distance 1-restricted pebbling in cycles and obtained the following result.

    Theorem 2.1. [13] Let Cn be an n-cycle, n3. Then

    π(1,t)(Cn)={nt2, if t is even and nt2 is even ;nt2+1, if t is even and nt2 is odd ;2n3, if t=1;nt2+n/(t+2)2, if t is odd and t3.

    In [13], Shiue also showed that if G is an r-regular graph of order n with girth at least 5 and t is a multiple of r2+r+4, then π(2,t)(G)=4ntr2+r+4. Since an n-cycle is 2-regular, we have the following.

    Corollary 2.1. [13] Let Cn be an n-cycle. If n5, then π(2,10k)(Cn)=4nk for each positive integer k.

    The conclusion of Corollary 2.1 gives the recurrence relation π(2,10k)(Cn)=π(2,10k10)(Cn)+4n=π(2,10k10)(Cn)+π(2,10)(Cn) for k2. Hence, we are interested in the following problem.

    Problem 1. For n5 and t11, determine the values of t that satisfy the recurrence relation

    π(2,t)(Cn)=π(2,t10)(Cn)+4n.

    It is easy to see that for any graph G and any two positive integers t1 and t2, π(2,t1+t2)(G)π(2,t1)(G)+π(2,t2)(G).

    Usually, the equality does not hold. For example, π(2,2)(C6)=6<8=π(2,1)(C6)+π(2,1)(C6).

    In this article, we mainly study the distance 2-restricted pebbling in cycles. Note that the diameter of C3 is equal to one. This implies that π(2,t)(C3)=πt(C3)=π(1,t)(C3), see Theorem 2.1. As C3=K3, an alternative result about πt(C3) can be seen in [12]. Regarding the research on πt(Cn), readers can refer to [14]. Note that the authors in [14] have only given an upper and a lower bound for πt(Cn) when n4 and t3. In this article, we show that for t13 and n6, π(2,t)(Cn)=π(2,t10)(Cn)+4n in Section 2 and completely determine the exact value of πt(Cn) for n=4,5, and t1 in Sections 4 and 5, respectively.

    Throughout the rest of this article, let Cn=v0,v1,,vn1,v0 be an n-cycle, n3, and all subscripts referring to the vertices of the cycle Cn will be interpreted modulo n. We write "a mod b" to denote the remainder when a is divided by b. For convenience, we use "[1,2]" to denote the set of all integers between 1 and 2 and, if δ is a distribution of pebbles on Cn, we use x0,x1,,xn1 to denote δ, where xi=δ(vi). Thus, |δ(Cn)|=n1i=0xi. Note that the diameter of Cn is not more than two for 3n5. It follows that π(2,t)(Cn)=πt(Cn) for 3n5. So we only consider the case when n6 in this section. For n6, and for i[0,n1],

    m(δ,2,vi)=12(xi22+xi1)+xi+12(xi+1+xi+22). (3.1)

    The following proposition is an important key for solving Problem 1.

    Proposition 3.1. For t11 and n6, if there exists an optimal (2,t)-solvable distribution δ=x0,x1,,xn1 of Cn such that xi4 for i[0,n1], then π(2,t)(Cn)=π(2,t10)(Cn)+4n.

    Proof. Assume that t11 and n6. By Corollary 2.1, we have π(2,10)(Cn)=4n. It follows that

    π(2,t)(Cn)π(2,t10)(Cn)+π(2,10)(Cn)=π(2,t10)(Cn)+4n.

    Thus,

    π(2,t10)(Cn)π(2,t)(Cn)4n.

    Assume that δ=x0,x1,,xn1 is an optimal (2,t)-solvable distribution of Cn with xi4 for all i[0,n1]. Then let δ=y0,y1,,yn1 be a distribution of Cn such that yi=xi4 for i[0,n1]. Clearly,

    |δ(Cn)|=|δ(Cn)|4n=π(2,t)(Cn)4n.

    To complete the proof, it is left to show that δ is (2,t10)-solvable. For i[0,n1], by (3.1), we have

    m(δ,2,vi)=yi22+yi12+yi+yi+22+yi+12=xi242+xi142+xi4+xi+242+xi+142=xi22+xi12+xi+xi+22+xi+1210=m(δ,2,vi)10t10.

    Thus, δ is (2,t10)-solvable, and we have the proof.

    Proposition 3.1 gives a sufficient condition for satisfying the recurrence relation in Problem 1. Now, we will find the values of t that satisfy the assumption of the statement in Proposition 3.1.

    Let δ=x0,x1,,xn1 be a (2,t)-solvable distribution of Cn with n6. Consider any subpath

    vj5,vj4,vj3,vj2,vj1,vj,vj+1,vj+2,vj+3,vj+4,vj+5

    in Cn, for some j[0,n1]. When 6n10, the subpath is the cycle Cn. We let n=10k, where 0k4. Then the vertex vj+5i=vj5+ki for i[0,k]. For example, if n=8, then vj+5=vj3, vj+4=vj4, and vj+3=vj5. By (3.1), we have

    m(δ,2,vj)=(xj2/2+xj1)/2+xj+(xj+1+xj+2/2)/2t.

    It follows that

    (xj2/2+xj1)+(xj+1+xj+2/2)2(txj).

    Without loss of generality, we can assume that

    xj2/2+xj1=t+rxj, (3.2)

    where r is a nonnegative integer. Then we have

    xj+1+xj+2/2trxj. (3.3)

    It follows that

    xj22(t+rxjxj1) (3.4)

    and

    xj+22(trxjxj+1). (3.5)

    By (3.1)–(3.5), we have the following four facts:

    Fact 1.

    m(δ,2,vj1)=(xj3/2+xj2)/2+xj1+(xj+xj+1/2)/2xj3/4+xj2/2+xj1+(xj+xj+1/2)/2=xj3/4+(t+rxj)+(xj+xj+1/2)/2.

    Fact 2.

    m(δ,2,vj+1)=(xj1/2+xj)/2+xj+1+(xj+2+xj+3/2)/2(xj1/2+xj)/2+xj+1+xj+2/2+xj+3/4(xj1/2+xj)/2+(trxj)+xj+3/4.

    Fact 3.

    m(δ,2,vj2)=(xj4/2+xj3)/2+xj2+(xj1+xj/2)/2(xj4/2+xj3)/2+2(t+rxjxj1)+(xj1+xj/2)/2.

    Fact 4.

    m(δ,2,vj+2)=(xj/2+xj+1)/2+xj+2+(xj+3+xj+4/2)/2(xj/2+xj+1)/2+2(trxjxj+1)+(xj+3+xj+4/2)/2.

    Fact 5. If xj28 and δ=y0,y1,,yn1 is a distribution of Cn such that yj=xj+2, yj1=xj1+2, yj2=xj28, yj3=xj3+4, and yi=xi for i[0,n1][j3,j], then |δ(Cn)|=|δ(Cn)|, m(δ,2,vj2)m(δ,2,vj2)5 and m(δ,2,vi)m(δ,2,vi) for i[0,n1]{j2}.

    Proof. Clearly, |δ(Cn)|=|δ(Cn)|. If n11, then the vertices

    vj5,vj4,vj3,vj2,vj1,vj,vj+1,vj+2,vj+3,vj+4,vj+5

    are all distinct. Otherwise, 6n10. Let n=10k, where 0k4. Then, the vertex vj+5i=vj5+ki for i[0,4]. By (3.1), it is easy to see that m(δ,2,vi)m(δ,2,vi) for i[0,n1]{j2,j1}. So, we only need to check m(δ,2,vj2) and m(δ,2,vj1). Since n6, vi{vj3,vj2,vj1,vj} for i{j4,j+1}. By (3.1), we have

    m(δ,2,vj2)=(yj4/2+yj3)/2+yj2+(yj1+yj/2)/2=(xj4/2+xj3+4)/2+xj28+(xj1+2+(xj+2)/2)/2(xj4/2+xj3)/2+xj2+(xj1+xj/2)/2+28+1=m(δ,2,vj2)5,

    and

    m(δ,2,vj1)=(yj3/2+yj2)/2+yj1+(yj+yj+1/2)/2=((xj3+4)/2+xj28)/2+xj1+2+(xj+2+xj+1/2)/2=(xj3/2+xj2)/2+xj1+(xj+xj+1/2)/23+2+1=m(δ,2,vj1).

    Fact 6. If xj24 and δ=y0,y1,,yn1 is a distribution of Cn such that yj=xj+1, yj1=xj1+1, yj2=xj24, yj3=xj3+2 and yi=xi for i[0,n1][j3,j], then |δ(Cn)|=|δ(Cn)|, m(δ,2,vj2)m(δ,2,vj2)3, m(δ,2,vj1)m(δ,2,vj1)1 and m(δ,2,vi)m(δ,2,vi) for i[0,n1]{j1,j2}.

    Proof. Clearly, |δ(Cn)|=|δ(Cn)|. If n11, then the vertices

    vj5,vj4,vj3,vj2,vj1,vj,vj+1,vj+2,vj+3,vj+4,vj+5

    are all distinct. Otherwise, 6n10. Let n=10k, where 0k4. Then the vertex vj+5i=vj5+ki for i[0,4]. By (3.1), it is easy to see that m(δ,2,vi)m(δ,2,vi) for i[0,n1]{j2,j1}. Now, we only need to check m(δ,2,vj2) and m(δ,2,vj1). Since n6, vi{vj3,vj2,vj1,vj} for i{j4,j+1}. By (3.1), we have

    m(δ,2,vj2)=(yj4/2+yj3)/2+yj2+(yj1+yj/2)/2=(xj4/2+xj3+2)/2+xj24+(xj1+1+(xj+1)/2)/2(xj4/2+xj3)/2+xj2+(xj1+xj)/2+14=m(δ,2,vj2)3,

    and

    m(δ,2,vj1)=(yj3/2+yj2)/2+yj1+(yj+yj+1/2)/2=((xj3+2)/2+xj24)/2+xj1+1+(xj+1+xj+1/2)/2(xj3/2+xj2)/2+xj1+(xj+xj+1/2)/22+1=m(δ,2,vj1)1.

    By using Eq (3.1) to check, we can obtain Facts 7 and 8.

    Fact 7. If xj14 and δ=y0,y1,,yn1 is a distribution of Cn such that yj=xj+2, yj1=xj14, yj2=xj2+2, and yi=xi for i[0,n1][j2,j], then |δ(Cn)|=|δ(Cn)|, m(δ,2,vj1)=m(δ,2,vj1)2 and m(δ,2,vi)m(δ,2,vi) for i[0,n1]{j1}.

    Fact 8. If xj12 and δ=y0,y1,,yn1 is a distribution of Cn such that yj=xj+1, yj1=xj12, yj2=xj2+1, and yi=xi for i[0,n1][j2,j], then |δ(Cn)|=|δ(Cn)|, m(δ,2,vj1)m(δ,2,vj1)2 and m(δ,2,vi)m(δ,2,vi) for i[0,n1]{j1}.

    Fact 9. If xj1 and xj+1 are both odd, and δ=y0,y1,,yn1 is a distribution of Cn such that yj=xj+2, yj1=xj11, yj+1=xj+11, and yi=xi for i[0,n1][j1,j,j+1], then |δ(Cn)|=|δ(Cn)| and m(δ,2,vi)m(δ,2,vi) for i[0,n1].

    Proof. Clearly, |δ(Cn)|=|δ(Cn)|. If n11, then the vertices

    vj5,vj4,vj3,vj2,vj1,vj,vj+1,vj+2,vj+3,vj+4,vj+5

    are all distinct. Otherwise, 6n10. Let n=10k, where 0k4. Then, the vertex vj+5i=vj5+ki for i[0,4]. By (3.1), it is easy to see that m(δ,2,vi)=m(δ,2,vi) for i[0,n1]{j3,j2,j1,j,j+1,j+2,j+3}. So, we only need to check m(δ,2,vi) for i{j3,j2,j1,j,j+1,j+2,j+3}. Note that xj1 and xj+1 are both odd. It follows that (xj11)/2=xj1/2 and (xj+11)/2=xj+1/2. Since n6, vi{vj1,vj,vj+1} for i{j4,j3,j2,j+2}. By using (3.1) to check, we can verify that m(δ,2,vi)m(δ,2,vi) for i{j,j1,j2}. For the value of m(δ,2,vj3), if n7, then vi{vj1,vj,vj+1} for i{j5,j4,j3,j2}, and

    m(δ,2,vj3)=(yj5/2+yj4)/2+yj3+(yj2+yj1/2)/2=(xj5/2+xj4)/2+xj3+(xj2+(xj11)/2)/2=(xj5/2+xj4)/2+xj3+(xj2+xj1/2)/2=m(δ,2,vj3).

    Otherwise, n=6, and hence vj5=vj+1; then we have

    m(δ,2,vj3)=(yj5/2+yj4)/2+yj3+(yj2+yj1/2)/2=((xj51)/2+xj4)/2+xj3+(xj2+(xj11)/2)/2=(xj5/2+xj4)/2+xj3+(xj2+xj1/2)/2=m(δ,2,vj3).

    By a similar argument as above, we also have m(δ,2,vi)m(δ,2,vi) for i{j+1,j+2,j+3}.

    Fact 10. If xj1 and xj are both odd, and δ=y0,y1,,yn1 is a distribution of Cn such that yj=xj+1, yj1=xj11, and yi=xi for i[0,n1][j1,j], then |δ(Cn)|=|δ(Cn)|, m(δ,2,vj1)m(δ,2,vj1)1 and m(δ,2,vi)m(δ,2,vi) for i[0,n1]{j1}.

    Proof. Clearly, |δ(Cn)|=|δ(Cn)|. If n11, then, the vertices

    vj5,vj4,vj3,vj2,vj1,vj,vj+1,vj+2,vj+3,vj+4,vj+5

    are all distinct. Otherwise, 6n10. Let n=10k, where 0k4. Then the vertex vj+5i=vj5+ki for i[0,4]. By (3.1), it is easy to see that m(δ,2,vi)=m(δ,2,vi) for i[0,n1]{j3,j2,j1,j,j+1,j+2}. So, we only need to check m(δ,2,vi) for i{j3,j2,j1,j,j+1,j+2}. Note that xj1 and xj are both odd. It follows that (xj11)/2=xj1/2 and (xj+1)/2=xj/2+1. Since n6, vi{vj1,vj} for i{j4,j3,j2,j+1,j+2}. By using (3.1) to check, we can verify that m(δ,2,vj1)m(δ,2,vj1)1 and m(δ,2,vj)m(δ,2,vj). For the value of m(δ,2,vj2), vi{vj1,vj} for i{j4,j3,j2}, and

    m(δ,2,vj2)=(yj4/2+yj3)/2+yj2+(yj1+yj/2)/2=(xj4/2+xj4)/2+xj2+(xj11+(xj+1)/2)/2=(xj4/2+xj3)/2+xj2+(xj1+xj/2)/2=m(δ,2,vj2).

    For the value of m(δ,2,vj3), vi{vj1,vj} for i{j5,j4,j3,j2}, and

    m(δ,2,vj3)=(yj5/2+yj4)/2+yj3+(yj2+yj1/2)/2=(xj5/2+xj4)/2+xj3+(xj2+(xj11)/2)/2=(xj5/2+xj4)/2+xj3+(xj2+xj1/2)/2=m(δ,2,vj3).

    By a similar argument as above, we also have m(δ,2,vi)m(δ,2,vi) for i{j+1,j+2}.

    Lemma 3.1. For t13 and n6, there exists an optimal (2,t)-solvable distribution δ of Cn such that δ(v)1 for each vertex v of Cn.

    Proof. Let δ=x0,x1,,xn1 be an optimal (2,t)-solvable distribution of Cn. Suppose that there exists a vertex vj, j[0,n1], with δ(vj)=xj=0. It suffices to show that there exists a (2,t)-solvable distribution δ of Cn such that δ(vj)1 and δ(vk)1 or δ(vk)xk for k[0,n1]{j} and |δ(Cn)||δ(Cn)|. By (3.1), we have

    (xj2/2+xj1)+(xj+1+xj+2/2)2(txj).

    Without loss of generality, we can assume that xj2/2+xj1=txj+r, where r is a nonnegative integer. This implies that (3.2)–(3.5) are valid. Now, we will modify the distribution of pebbles on the vertices in {vi|i[j3,j+3]} if necessary. Let δ=y0,y1,,yn1 be a distribution of Cn such that yi=xi for i[0,n1][j3,j+3]. For i[j3,j+3], yi will be defined according to the following cases. Initially, we let yi=xi for i[j3,j+3].

    By (3.4) and Fact 3, we have

    xj22(t+rxjxj1)=2(13+rxj1)

    and

    m(δ,2,vj2)(xj4/2+xj3)/2+2(t+rxjxj1)+(xj1+xj/2)/2t+13+2r(2xj1xj1/2).

    Case 1. r=0.

    If xj15, then xj22(13+05)=16 and m(δ,2,vj2)t+13+0(255/2)=t+5. Let yj=xj+2=2, yj1=xj1+2, yj2=xj288, and yj3=xj3+4. By Fact 5, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    Similarly, if xj+15, then, by (3.5) and Fact 4, we have xj+216 and m(δ,2,vj+2)t+5. Let yj=xj+2=2, yj+1=xj+1+2, yj+2=xj+288, yj+3=xj+3+4. Then, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    Now, we can assume that xj16 and xj+16. Let yj=xj+1=1, yj1=xj124, and yj2=xj2+1. By Fact 8, we only need to check m(δ,2,vj1). By (3.1) and (3.2), we have

    m(δ,2,vj1)=(yj3/2+yj2)/2+yj1+(yj+yj+1/2)/2xj3/4+(xj2+1)/2+xj12+(xj+1+xj+1/2)/2(txj+r)2+(1+6/2)/2=t.

    Case 2. r=1.

    If xj16, then xj22(13+16)=16 and m(δ,2,vj2)t+13+2(266/2)=t+6. Let yj=xj+2=2, yj1=xj1+2, yj2=xj288, and yj3=xj3+4. Then, by Fact 5, we have |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. Otherwise, xj17. If xj+14, then, by Fact 1, we have m(δ,2,vj1)(t+rxj)+(xj+xj+1/2)/2t+2. Let yj=xj+2=2, yj1=xj143, and yj2=xj2+2. Then, by Fact 7, we have |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. If xj+13, then, by (3.5) and Fact 4, xj+22(trxjxj+1)2(13103)=18 and m(δ,2,vj+2)xj+1/2+2(trxjxj+1)3/2+t+13+2(103)=t+6. Let yj=xj+2=2, yj+1=xj+1+2, yj+2=xj+2810, and yj+3=xj+3+4. Then, by Fact 5, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    Case 3. r2.

    If xj18, then xj22(13+28)=14 and m(δ,2,vj2)t+13+4(288/2)+4=t+5. Let yj=xj+2=2, yj1=xj1+2, yj2=xj286, and yj3=xj3+4. Then, by Fact 5, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. Otherwise, xj19. By Fact 1, m(δ,2,vj1)t+rxjt+2. Let yj=xj+2=2, yj1=xj145, yj2=xj2+2. Then, by Fact 7, we have |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    Lemma 3.2. For t13 and n6, there exists an optimal (2,t)-solvable distribution δ of Cn such that δ(v)2 for each vertex v of Cn.

    Proof. By Lemma 3.1, we assume that δ=x0,x1,,xn1 is an optimal (2,t)-solvable distribution of Cn such that δ(v)1 for each vertex v of Cn. Suppose that xj=1 for some j[0,n1]. Then, let δ=y0,y1,,yn1 be defined as the description in Lemma 3.1. It suffices to show that δ(vj)2 and δ(vk)2 or δ(vk)xk for k[0,n1]{j} and |δ(Cn)||δ(Cn)|.

    By (3.4) and Fact 3, we have xj22(12+rxj1) and m(δ,2,vj2)t+11+2r(2xj1xj1/2).

    Case 1. r=0.

    If xj14, then xj22(12+04)=16 and m(δ,2,vj2)t+11+0(244/2)=t+5. Let yj=xj+2=3, yj1=xj1+2, yj2=xj288, and yj3=xj3+4. By Fact 5, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    Similarly, if xj+14, then xj+216 and m(δ,2,vj+2)t+5. Let yj=xj+2=3, yj+1=xj+1+2, yj+2=xj+288, yj+3=xj+3+4. Then, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    Now, we assume that xj15 and xj+15. Note that xj=1 is odd. If xj1 is odd, let yj=xj+1=2, and yj1=xj114. By Fact 10, we only need to check m(δ,2,vj1). By using (3.1) and (3.2), we can verify m(δ,2,vj1)t.

    Similarly, if xj+1 is odd, let yj=xj+1=2, and yj+1=xj+114. Then, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    Now, we can assume that xj16 and xj+16, and xj1 and xj+1 are both even. For the case xj16 and xj+18, let yj=xj+1=2, yj1=xj124, and yj2=xj2+1. Then, by Fact 8, we only need to check m(δ,2,vj1). Also, by using (3.1) and (3.2), it is easy to check that m(δ,2,vj1)t.

    Similarly, for the case xj18 and xj+16, let yj=xj+1=2, yj+1=xj+124, and yj+2=xj+2+1. Then, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    For the case xj1=xj+1=6, let yj=xj+1=2, yj1=xj11=5, yj2=xj212(t+rxjxj1)12(13+016)111, and yj3=xj3+1. Clearly, |δ(Cn)|=|δ(Cn)|. If n11, then the vertices

    vj5,vj4,vj3,vj2,vj1,vj,vj+1,vj+2,vj+3,vj+4,vj+5

    are all distinct. Otherwise, 6n10. Let n=10k, where 0k4. Then the vertex vj+5i=vj5+ki for i[0,4]. By (3.1), it is easy to see that m(δ,2,vi)=m(δ,2,vi) for i[0,n1]{j5,j4,j3,j2,j1,j,j+1,j+2}. If n9, vi{vj3,vj2,vj1,vj} for i{j7,j6,j5,j4,j+1,j+2,j+3,j+4,j+5}. Note that when n=9, vj+5=vj4 and vj+4=vj5, or when n=10, vj+5=vj5. So, for n9, it is easy to see that m(δ,2,vi)m(δ,2,vi) for i{j5,j4,j+1,j+2}. By (3.1)–(3.3), we also can verify that m(δ,2,vi)t for i{j,j1,j2,j3}. For the case that 6n8, without loss of generality, let j=0. Then, by (3.1)–(3.3), it is easy to verify that δ is (2,t)-solvable by checking m(δ,2,vi) for each i[0,n1].

    Case 2. r=1.

    If xj15, then xj22(12+15)=16 and m(δ,2,vj2)t+11+2(255/2)=t+5. Let yj=xj+2=3, yj1=xj1+2, yj2=xj288, and yj3=xj3+4. By Fact 5, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. If xj+12, then, by (3.5) and Fact 4, xj+22(trxjxj+1)2(13112)=18 and m(δ,2,vj+2)xj+1/2+2(trxjxj+1)2/2+t+132(1+1+2)=t+6. Let yj=xj+2=3, yj+1=xj+1+2, yj+2=xj+2810, and yj+3=xj+3+4. Then, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. Now, we can assume that xj16 and xj+13. If xj+14, let yj=xj+1=2, yj1=xj124, and yj2=xj2+1. By Fact 8, we only need to check m(δ,2,vj1). By using (3.1) and (3.2), we can verify that m(δ,2,vj1)t. For the case xj16 and xj+1=3, let yj=xj+1=2, yj1=xj124, yj2=xj2+1, yj+1=xj+1+1, yj+2=xj+222(13113)214 and yj+3=xj+3+1. By Fact 8, we only need to check m(δ,2,vj1) and m(δ,2,vj+2). By using (3.1), (3.2) and (3.5), we can verify that m(δ,2,vj1)t and m(δ,2,vj+2)t+4.

    Case 3. r2.

    If xj16, then xj22(12+26)=16 and m(δ,2,vj2)t+11+4(266/2)=t+6. Let yj=xj+2=3, yj1=xj1+2, yj2=xj288, and yj3=xj3+4. Then, by Fact 5, we have |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. Otherwise, xj17. Let yj=xj+1=2, yj1=xj125, and yj2=xj2+1. Then, by Fact 8, we only need to check m(δ,2,vj1). By using (3.1) and (3.2), we can verify that m(δ,2,vj1)t.

    Lemma 3.3. For t13 and n6, there exists an optimal (2,t)-solvable distribution δ of Cn such that δ(v)3 for each vertex v of Cn.

    Proof. By Lemma 3.2, we assume that δ=x0,x1,,xn1 is an optimal (2,t)-solvable distribution of Cn such that δ(v)2 for each vertex v of Cn. Suppose that xj=2 for some j[0,n1]. Then, let δ=y0,y1,,yn1 be defined as the description in Lemma 3.1. It suffices to show that δ(vj)3 and δ(vk)3 or δ(vk)xk for k[0,n1]{j} and |δ(Cn)||δ(Cn)|.

    By (3.4) and Fact 3, we have xj22(11+rxj1) and m(δ,2,vj2)t+10+2r(2xj1(xj1+1)/2).

    Case 1. r=0.

    If xj13, then xj22(11+03)=16 and m(δ,2,vj2)t+10+0(23(3+1)/2)=t+6. Let yj=xj+2=4, yj1=xj1+2, yj2=xj288, and yj3=xj3+4. Similarly, if xj+13, then xj+216 and m(δ,2,vj+2)t+6. Let yj=xj+2=4, yj+1=xj+1+2, yj+2=xj+288, and yj+3=xj+3+4. By Fact 5, we have |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    Now, we can assume that xj14 and xj+14. For the case xj1=4 and xj+14, we have xj22(11+04)=14 and m(δ,2,vj2)t+10+0(24(4+1)/2)=t+4. Let yj=xj+1=3, yj1=xj1+1, yj2=xj2410, and yj3=xj3+2. Then, by Fact 6, we only need to check m(δ,2,vj1). By (3.1) and (3.2), we can verify m(δ,2,vj1)t.

    Similarly, for the case xj14 and xj+1=4, let yj=xj+1=3, yj+1=xj+1+1, yj+2=xj+2410, and yj+3=xj+3+2. Then, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. Now, we can assume that xj15 and xj+15. For the case xj1=xj+1=5, let yj=xj+2=4, yj1=xj11=4, and yj+1=xj+11=4. Note that xj1 and xj+1 are both odd. By Fact 9, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. For the case xj15 and xj+16, let yj=xj+1=3, yj1=xj123, and yj2=xj2+1. Then, by Fact 8, we only need to check m(δ,2,vj1). By using (3.1) and (3.2), we can verify m(δ,2,vj1)t.

    Similarly, for the case xj16 and xj+15, let yj=xj+1=3, yj+1=xj+123, and yj+2=xj+2+1. Then, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable.

    Case 2. r=1.

    If xj14, then xj22(11+14)=16 and m(δ,2,vj2)t+10+2(244/2)=t+6. Let yj=xj+2=4, yj1=xj1+2, yj2=xj288, and yj3=xj3+4. Then, by Fact 5, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. Otherwise, xj15, let yj=xj+1=3, yj1=xj123, and yj2=xj2+1. By Fact 8, we only need to check m(δ,2,vj1). By (3.1) and (3.2), we can verify m(δ,2,vj1)t.

    Case 3. r2.

    If xj16, then xj22(11+26)=14 and m(δ,2,vj2)t+10+4(266/2)=t+5. Let yj=xj+2=4, yj1=xj1+2, yj2=xj286, and yj3=xj3+4. Then, by Fact 5, |δ(Cn)|=|δ(Cn)| and δ is (2,t)-solvable. Otherwise, xj17, let yj=xj+1=3, yj1=xj125, and yj2=xj2+1. By Fact 8, we only need to check m(\delta', 2, v_{j-1}) . By (3.1) and (3.2), we can verify m(\delta', 2, v_{j-1})\ge t+1 .

    Lemma 3.4. For t\ge 13 and n\ge 6 , there exists an optimal (2, t) -solvable distribution \delta of C_n such that \delta(v)\ge 3 for each vertex v of C_n and x_k = 3 implies x_{k+1}\ge 4 for any k\in [0, n-1] .

    Proof. By Lemma 3.3, we assume that \delta = \langle x_0, x_1, \cdots, x_{n-1}\rangle is an optimal (2, t) -solvable distribution of C_n such that \delta(v)\ge 3 for each vertex v of C_n . Suppose that x_{j-1} = 3 and x_j = 3 or x_{j} = 3 and x_{j+1} = 3 for some j\in [0, n-1] . Then, let \delta' = \langle y_0, y_1, \cdots, y_{n-1}\rangle be defined as the description in Lemma 3.1. For the case that x_{j-1} = 3 and x_j = 3 , it suffices to show that \delta'(v_k)\ge 4 for k = j or j-1 and \delta'(v_i)\ge 4 or \delta'(v_i) = x_i for i\in [0, n-1]\backslash\{k\} and |\delta'(C_n)|\leq|\delta(C_n)| . For the case that x_{j} = 3 and x_{j+1} = 3 , it suffices to show that \delta'(v_k)\ge 4 for k = j or j+1 and \delta'(v_i)\ge 4 or \delta'(v_i) = x_i for i\in [0, n-1]\backslash\{k\} and |\delta'(C_n)|\leq|\delta(C_n)| .

    Case 1. x_{j-1} = 3 and x_j = 3 .

    By (3.4) and Fact 3, we have

    \begin{align*} x_{j-2}&\ge 2(t+r-x_j-x_{j-1})\\ & = 2(13+r-3-3)\\ &\ge 14 \end{align*}

    and

    \begin{align*} m(\delta, 2, v_{j-2})&\ge \lfloor (\lfloor x_{j-4}/2\rfloor +x_{j-3})/2 \rfloor +2(t+r-x_j-x_{j-1}) + \lfloor (x_{j-1}+\lfloor x_j/2\rfloor )/2 \rfloor\\ &\ge\lfloor (\lfloor 3/2\rfloor +3)/2 \rfloor+ t+13+2r-6-6 +\lfloor(3+\lfloor 3/2 \rfloor)/2 \rfloor\\ &\ge t+5 . \end{align*}

    Let y_j = x_j+2 = 5 , y_{j-1} = x_{j-1}+2 = 5 , y_{j-2} = x_{j-2}-8\ge 6 , and y_{j-3} = x_{j-3}+4 . Then, by Fact 5, \delta' is a desired distribution.

    Case 2. x_{j} = 3 and x_{j+1} = 3 .

    If r = 0 , then by (3.5) and Fact 4, we have x_{j+2}\ge 14 and m(\delta, 2, v_{j+2})\ge t+5 . Let y_j = x_j+2 = 5 , y_{j+1} = x_{j+1}+2 = 5 , y_{j+2} = x_{j+2}-8\ge 6 , and y_{j+3} = x_{j+3}+4\ge 7 . Then, by Fact 5, \delta' is a desired distribution. Otherwise, r\ge 1 . Suppose that r = 1 . Then, by (3.5) and Fact 4, we have x_{j+2}\ge 12 and m(\delta, 2, v_{j+2})\ge t+3 . Let y_{j+1} = x_{j+1}+2 = 5 , y_{j+2} = x_{j+2}-4\ge 8 , and y_{j+3} = x_{j+3}+2\ge 5 . Then, by Fact 7, \delta' is a desired distribution. Suppose that r\ge 2 and x_{j-1}\le 5 . Then, by (3.4) and Fact 3, we have x_{j-2}\ge 14 and m(\delta, 2, v_{j-2})\ge t+5 . Let y_j = x_j+2 = 5 , y_{j-1} = x_{j-1}+2\ge 5 , y_{j-2} = x_{j-2}-8\ge 6 , and y_{j-3} = x_{j-3}+4\ge 7 . Then, by Fact 5, \delta' is a desired distribution. Suppose that r\ge 2 and x_{j-1}\ge 6 . Then, by Fact 1, we have m(\delta, 2, v_{j-1})\ge t+1 . If x_{j-1} is odd, let y_j = x_j+1 = 4 , y_{j-1} = x_{j-1}-1\ge 5 . We only need to check m(\delta', 2, v_{j-2}) and m(\delta', 2, v_{j-3}) . By using (3.1), it is easy to see that m(\delta', 2, v_{j-2}) = m(\delta, 2, v_{j-2}) and m(\delta', 2, v_{j-3}) = m(\delta, 2, v_{j-3}) . Otherwise, x_{j-1} is even, let y_j = x_j+1 = 4 , y_{j-1} = x_{j-1}-2\ge 4 , and y_{j-2} = x_{j-2}+1 \ge 4 . We only need to check m(\delta', 2, v_{j-1}) . By (3.1) and (3.2), we have

    \begin{align*} \ m(\delta', 2, v_{j-1})& = \lfloor (\lfloor y_{j-3}/2\rfloor +y_{j-2})/2 \rfloor +y_{j-1}+ \lfloor (y_j+\lfloor y_{j+1}/2\rfloor )/2 \rfloor\\ & = \lfloor (\lfloor x_{j-3}/2\rfloor +x_{j-2}+1)/2 \rfloor +x_{j-1}-2+ \lfloor (x_j+1+\lfloor x_{j+1}/2\rfloor )/2 \rfloor\\ &\ge \lfloor(\lfloor 3/2\rfloor+1)/2\rfloor+\lfloor x_{j-2}/2\rfloor+x_{j-1}-2+ \lfloor (4+\lfloor x_{j+1}/2\rfloor )/2 \rfloor\\ &\ge 1+(t+r-x_j)-2+2 \\ & = t. \end{align*}

    This completes the proof.

    We demonstrate the techniques of Lemmas 3.1–3.4 with the following example.

    Let \delta = \langle 4, 8, 12, 0, 3, 14\rangle be a distribution of C_6 such that 0 = x_j . Note that \delta is (2, 13) -solvable. By Eq (3.2), r = 3 . Since x_{j-1}\ge 9 , by Case 3 in Lemma 3.1, \delta is modified to \langle 4, 10, 8, 2, 3, 14 \rangle with x_j = 2 . Note that at this stage, the new \delta is still (2, 13) -solvable, \delta(v)\ge 2 for all v\in V(C_6) and r = 2 . Since x_{j-1}\ge 7 , by Case 3 of Lemma 3.3, \delta is modified to \langle 4, 11, 6, 3, 3, 14 \rangle . Note that at this stage the new \delta is still (2, 13) -solvable, \delta(v)\ge 3 for all v\in V(C_6) , x_j = x_{j+1} = 3 , and r = 1 . By Case 2 of Lemma 3.4, \delta is modified to \langle 6, 11, 6, 3, 5, 10 \rangle . Note that x_{j+1}\ge 4 and \delta is still (2, 13) -solvable. Now, we are in a position to make a (2, 13) -solvable \delta such that \delta(v)\ge 4 for each vertex v of the cycle with the following proposition.

    Proposition 3.2. If t\ge 13 and n\ge 6 , then there exists an optimal (2, t) -solvable distribution \delta of C_n such that \delta(v)\ge 4 for each vertex v of C_n .

    Proof. By Lemma 3.4, we assume that \delta = \langle x_0, x_1, \cdots, x_{n-1}\rangle is an optimal (2, t) -solvable distribution of C_n such that x_i\ge 3 for all i\in [0, n-1] and x_k = 3 implies x_{k+1}\ge 4 for any k\in [0, n-1] . Suppose that x_j = 3 for some j\in [0, n-1] . Then, let \delta' = \langle y_0, y_1, \cdots, y_{n-1}\rangle be defined as the description in Lemma 3.1. It suffices to show that \delta'(v_j)\ge 4 and \delta'(v_k)\ge 4 or \delta'(v_k) = x_k for k\in [0, n-1]\backslash\{j\} and |\delta'(C_n)|\leq|\delta(C_n)| .

    By (3.4) and Fact 3, we have x_{j-2}\ge 2(10+r-x_{j-1}) and m(\delta, 2, v_{j-2})\ge t+9+2r-(2x_{j-1}- \lfloor (x_{j-1}+1)/2 \rfloor) .

    Case 1. r = 0 .

    If 4 \le x_{j-1}\le 6 and x_{j+1}\ge 4 , let y_j = x_j+1 = 4 , y_{j-2} = x_{j-2}-2\ge 2(t+r-x_j-x_{j-1})-2\ge 2(13+0-3-6)-2\ge 6 , and y_{j-3} = x_{j-3}+1 . Clearly, |\delta'(C_n)| = |\delta(C_n)| . By (3.1), it is easy to see that m(\delta', 2, v_i) = m(\delta, 2, v_i) for i\in [0, n-1]\setminus \{j-5, j-4, j-3, j-2, j-1, j, j+1, j+2\} . For n\ge 9 , v_i\notin \{v_{j-3}, v_{j-2}, v_j\} for i\in \{j-7, j-6, j-5, j-4, j+1, j+2, j+3, j+4, j+5\} . By (3.1), it is easy to check that m(\delta', 2, v_i)\ge m(\delta, 2, v_i) for i\in \{j-5, j-4, j-3, j, j+1, j+2\} . Hence, we only need to check m(\delta', 2, v_{j-1}) and m(\delta', 2, v_{j-2}) . Note that x_{j-3}\ge 4 or x_{j-4}\ge 4 . It follows that \lfloor x_{j-4}/2\rfloor+x_{j-3}\ge 5 . By (3.1)–(3.3), we have

    \begin{align*} m(\delta', 2, v_{j-1})& = \lfloor (x_{j-3}+1)/4\rfloor+\lfloor (x_{j-2}-2)/2\rfloor+x_{j-1} + \lfloor (x_j+1+\lfloor x_{j+1}/2\rfloor )/2 \rfloor\\ &\ge \lfloor (3+1)/4\rfloor+ (t+r-x_j)-1+ \lfloor (4+\lfloor 4/2\rfloor )/2 \rfloor\\ & = t, \end{align*}

    and

    \begin{align*} m(\delta', 2, v_{j-2}) & = \lfloor(\lfloor x_{j-4}/2\rfloor+x_{j-3}+1)/2\rfloor+x_{j-2}-2 + \lfloor (x_{j-1}+\lfloor (x_j+1)/2\rfloor )/2 \rfloor\\ &\ge \lfloor (5+1)/2\rfloor+2(t+r-x_j-x_{j-1})-2 +\lfloor(x_{j-1}+\lfloor 4/2\rfloor )/2 \rfloor\\ &\ge t. \end{align*}

    For the case that 6\le n \le 8 , without loss of generality, let j = 0 . Then by (3.1)–(3.3), it is easy to verify that \delta' is (2, t) -solvable by checking m(\delta', 2, v_i) for each i\in [0, n-1] .

    Similarly, if x_{j-1}\ge 4 and 4\le x_{j+1}\le 6 , let y_j = x_j+1 = 4 , y_{j+2} = x_{j+2}-2\ge 6 , and y_{j+3} = x_{j+3}+1 . Then, |\delta'(C_n)| = |\delta(C_n)| and \delta' is (2, t) -solvable.

    Now, we can assume that x_{j-1}\ge 7 and x_{j+1}\ge 7 . If x_{j-1} = x_{j+1} = 7 , then x_{j-1} and x_{j+1} are both odd. Let y_j = x_j+2 = 5 , y_{j-1} = x_{j-1}-1 = 6 and y_{j+1} = x_{j+1}-1 = 6 . Then, by Fact 9, we have |\delta'(C_n)| = |\delta(C_n)| and \delta' is (2, t) -solvable.

    For the case x_{j-1}\ge 7 and x_{j+1}\ge 8 , let y_j = x_j+1 = 4 , y_{j-1} = x_{j-1}-2\ge 5 , and y_{j-2} = x_{j-2}+1 . Then, by Fact 8, we only need to check m(\delta', 2, v_{j-1}) . By (3.1) and (3.2), we can verify m(\delta', 2, v_{j-1})\ge t .

    Similarly, for the case x_{j-1}\ge 8 and x_{j+1}\ge 7 , let y_j = x_j+1 = 4 , y_{j+1} = x_{j+1}-2\ge 5 , and y_{j+2} = x_{j+2}+1 . Then, |\delta'(C_n)| = |\delta(C_n)| and \delta' is (2, t) -solvable.

    Case 2. r = 1 .

    If x_{j-1} = 4 , then x_{j-2}\ge 2(10 +1-4) = 14 and m(\delta, 2, v_{j-2})\ge t+9+2-(2\cdot 4-\lfloor 4/2 \rfloor) = t+5 . Let y_j = x_j+2 = 5 , y_{j+1} = x_{j+1}+2 , y_{j-2} = x_{j-2}-8\ge 6 , and y_{j-3} = x_{j-3}+4 . By Fact 5, |\delta'(C_n)| = |\delta(C_n)| and \delta' is (2, t) -solvable. Otherwise, x_{j-1}\ge 5 . If x_{j-1} = 5 , then x_{j} and x_{j-1} are both odd. Let y_j = x_j+1 = 4 and y_{j-1} = x_{j-1}-1 = 4 . By Fact 10, we only need to check m(\delta', 2, v_{j-1}) . By (3.1) and (3.2), we can verify m(\delta', 2, v_{j-1})\ge t .

    Now, we can assume that x_{j-1}\ge 6 . Let y_j = x_j+1 = 4 , y_{j-1} = x_{j-1}-2\ge 4 , and y_{j-2} = x_{j-2}+1 . By Fact 8, we only need to check m(\delta', 2, v_{j-1}) . By using (3.1) and (3.2), we can verify m(\delta', 2, v_{j-1})\ge t .

    Case 3. r\ge 2 .

    If x_{j-1}\le 5 , then x_{j-2}\ge 2(10+2-5) = 14 and m(\delta, 2, v_{j-2})\ge t+9+4-(2\cdot 5-\lfloor 5/2 \rfloor) = t+5 . Let y_j = x_j+2 = 5 , y_{j-1} = x_{j-1}+2 , y_{j-2} = x_{j-2}-8\ge 6 , and y_{j-3} = x_{j-3}+4 . By Fact 5, we have |\delta'(C_n)| = |\delta(C_n)| and \delta' is (2, t) -solvable. Otherwise, x_{j-1}\ge 6 . Let y_j = x_j+1 = 4 , y_{j-1} = x_{j-1}-2\ge 4 , and y_{j-2} = x_{j-2}+1 . By Fact 8, we only need to check m(\delta', 2, v_{j-1}) . By using (3.1) and (3.2), we can verify m(\delta', 2, v_{j-1})\ge t+1 .

    As a demonstration of the techniques used in the proof of Proposition 3.2, we give one example for each of the three cases.

    For t = 13, n = 6 , we let \delta = \langle x_{j-3}, x_{j-2}, x_{j-1}, x_j, x_{j+1}, x_{j+2}\rangle .

    If \delta = \langle 4, 12, 4, 3, 4, 12\rangle , then, \delta is a (2, 13) -solvable distribution of C_6 with x_j = 3 . By Eq (3.2), r = 0 . Since x_{j-1} = x_{j+1} = 4 , \delta' = \langle 5, 10, 4, 4, 4, 12\rangle .

    If \delta = \langle 6, 11, 6, 3, 5, 10\rangle , then \delta is (2, 13) -solvable and r = 1 . Since x_{j-1}\ge 6 , \delta' = \langle 6, 12, 4, 4, 5, 10\rangle . Note that this is a continuation of the example above Proposition 3.2.

    If \delta = \langle 4, 12, 6, 3, 4, 12\rangle , then \delta is (2, 13) -solvable and r = 2 . Since x_{j-1}\ge 6 , \delta' = \langle 4, 13, 4, 4, 4, 12\rangle .

    Note that in all examples above \delta'(v)\ge 4 for each v\in V(C_6) and \delta' is still (2, 13) -solvable.

    By combining Propositions 3.1 and 3.2, we have the following.

    Theorem 3.1. For t\ge 13 and n\ge 6 , \pi_{(2, t)}^*(C_n) = \pi_{(2, t-10)}^*(C_n)+4n .

    By using Theorem 3.1 repeatedly (if necessary), we have the following.

    Corollary 3.1. For n\ge 6 , \pi^*_{(2, 10k+r)}(C_n) = \pi^*_{(2, r)}(C_n)+4kn , where k\ge 1 and 3\le r \le 12 .

    Let \delta = \langle x_0, x_1, x_2, x_3 \rangle be a (2, t) -solvable distribution of C_4 . Note that v_{i-2} = v_{i+2} for all i\in [0, 3] . Hence, Eq (3.1) is not true for C_4 ; it must be modified into the following:

    \begin{equation} m(\delta, 2, v_i) = \max\{m_i | y_{i-2}+y_{i+2} = x_{i+2}\}, \end{equation} (4.1)

    where m_i = \left\lfloor (\lfloor y_{i-2}/2\rfloor+x_{i-1})/2\right\rfloor+x_i+\left \lfloor (x_{i+1}+\lfloor y_{i+2}/2\rfloor)/2\right \rfloor .

    This implies that if x_{i+2}\le 1 , then

    m(\delta, 2, v_i) = \left\lfloor \frac{x_{i-1}}{2}\right\rfloor+x_i+\left \lfloor \frac{x_{i+1}}{2}\right \rfloor+\lfloor \frac{x_{i+2}}{4}\rfloor.

    Otherwise,

    m(\delta, 2, v_i) = x_i+\left \lfloor \frac{1}{2}(x_{i-1}+x_{i+1}+\lfloor \frac{x_{i+2}}{2}\rfloor)\right \rfloor.

    It follows that \frac{9}{4}(x_0+x_1+x_2+x_3)\ge4t or x_0+x_1+x_2+x_3\ge\frac{16t}{9} . This implies

    \begin{equation} \pi^*_{(2, t)}(C_4)\ge\left\lceil \frac{16t}{9}\right \rceil. \end{equation} (4.2)

    It is not difficult to see \langle 4k, 4k, 4k, 4k \rangle is an optimal (2, 9k) -solvable distribution of C_4 for k\ge 1 . Thus, we have the following.

    Proposition 4.1. For k\ge 1 , \pi^*_{(2, 9k)}(C_4) = 16k .

    Clearly, \pi^*_{(2, 9k+r)}(C_4)\le \pi^*_{(2, 9k)}(C_4)+\pi^*_{(2, r)}(C_4) for r\ge 1 . Also, by Proposition 4.1, for k\ge 1 , we have

    \begin{equation} \pi^*_{(2, 9k+r)}(C_4)\le 16k+\pi^*_{(2, r)}(C_4). \end{equation} (4.3)

    It is known that \pi^*(C_4) = \lceil \frac{2\times 4}{3}\rceil = 3 , see [2]. So, we have the following.

    Proposition 4.2. \pi^*_{(2, 1)}(C_4) = 3 .

    Proposition 4.3. If t = 9k+1 and k\ge 1 , then \pi^*_{(2, t)}(C_4) = \lceil 16t/9\rceil .

    Proof. For k = 1 , by (4.2), we have \pi^*_{(2, 10)}(C_4)\ge\lceil 160/9\rceil = 18 . By (4.1), it is easy to check that \langle 4, 5, 4, 5 \rangle is a (2, 10) -solvable distribution of C_4 . Hence, we have \pi^*_{(2, 10)}(C_4) = 18 . For k\ge 2 , by (4.2), we have \pi^*_{(2, 9k+1)}(C_4)\ge\lceil 16(9k+1)/9\rceil . Also, by (4.3), we have \pi^*_{(2, 9k+1)}(C_4) = \pi^*_{(2, 9(k-1)+10)} (C_4)\le 16(k-1)+\pi^*_{(2, 10)} (C_4) = 16(k-1)+18 = \lceil 16(9k+1)/9\rceil . This completes the proof.

    Proposition 4.4. For r\in \{2, 3, 4, 6, 7, 8\} , if t = 9k+r and k\ge 0 , then \pi^*_{(2, t)}(C_4) = \lceil 16t/9\rceil .

    Proof. For k\ge 0 and r\ge 0 , by (4.2), we have \pi^*_{(2, 9k+r)}(C_4)\ge\lceil 16(9k+r)/9\rceil = 16k+\lceil \frac{16r}{9}\rceil . Also, by (4.3), we have \pi^*_{(2, 9k+r)}(C_4)\le 16k+\pi^*_{(2, r)} (C_4) . So, it suffices to prove that \pi^*_{(2, r)}(C_4)\le \lceil \frac{16r}{9}\rceil for r\in \{2, 3, 4, 6, 7, 8\} . We will prove it by constructing a (2, r) -solvable distribution with \lceil \frac{16r}{9}\rceil pebbles for r\in \{2, 3, 4, 6, 7, 8\} . Let

    \begin{align*} \delta_2 = \langle 2, 0, 2, 0 \rangle,\; \delta_3 = \langle 2, 0, 4, 0 \rangle,\; \delta_4 = \langle 2, 2, 2, 2 \rangle, \\ \delta_6 = \langle 4, 2, 3, 2 \rangle,\; \delta_7 = \langle 4, 3, 3, 3 \rangle,\; \delta_8 = \langle 4, 4, 3, 4 \rangle. \end{align*}

    It is easy to see |\delta_r(C_5)| = \lceil \frac{16r}{9}\rceil and by using (4.1), it is not difficult to verify that \delta_r is a (2, r) -solvable distribution of C_4 for each r\in \{2, 3, 4, 6, 7, 8\} .

    Proposition 4.5. If t = 9k+5 and k\ge 0 , then \pi^*_{(2, t)}(C_4) = \lceil 16t/9\rceil+1 .

    Proof. For k = 0 , by (4.1), it is easy to check that \langle 4, 2, 2, 2 \rangle is a (2, 5) -solvable distribution of C_4 . Hence, we have \pi^*_{(2, 5)}(C_4)\le 10 . For k\ge 1 , by (**2) , we have \pi^*_{(2, 9k+5)}(C_4)\le 16k+\pi^*_{(2, 5)} (C_4)\le 16k+10 . Assume that \delta = \langle x_0, x_1, x_2, x_3 \rangle is a (2, 9k+5) -solvable distribution of C_4 with x_0+x_1+ x_2+x_3 = 16k+9 , where k\ge 0 . Let x_i = 4k+r_i , i = 0, 1, 2, 3 . Then, r_0+r_1+r_2+r_3 = 9 . Without loss of generality, let r_0 = \min \{r_i|i = 0, 1, 2, 3\} . By (4.1), we have x_0+(x_1+x_3)/2+x_2/4 = 9k+r_0+(r_1+r_3)/2+r_2/4\ge m(\delta, 2, v_0)\ge 9k+5 . This implies that r_0+(r_1+r_3)/2+r_2/4\ge 5 , equivalently, r_0+r_1+r_2+r_3\ge 10-r_0+r_2/2 . Thus, 9\ge 10-r_0+r_2/2 , and we have r_0\ge 1 + r_2/2\ge 1 + r_0/2 . Hence, r_0 \ge 2 . This implies that \delta is one of the following distributions:

    \begin{align*} \langle 4k+2,4k+3,4k+2,4k+2 \rangle, \\ \langle 4k+2,4k+2,4k+3,4k+2 \rangle, \\ \langle 4k+2,4k+2,4k+2,4k+3 \rangle. \end{align*}

    It is easy to check that \delta is not (2, 9k+5) -solvable, which is a contradiction. Therefore, \pi^*_{(2, 9k+5)}(C_4)\ge 16k+10 = \lceil 16(9k+5)/9\rceil +1 , and we have the proof.

    By Propositions 4.1–4.5, we conclude the following.

    Theorem 4.1. Let t be a positive integer. If t = 1 or t\mod 9 = 5 , then \pi_{(2, t)}^*(C_4) = \lceil 16t/9\rceil+1 . Otherwise, \pi_{(2, t)}^*(C_4) = \lceil 16t/9\rceil .

    Let \delta = \langle x_0, x_1, x_2, x_3, x_4 \rangle be a (2, t) -solvable distribution of C_5 . Then we have

    \begin{equation} x_i+(x_{i+1}+x_{i-1})/2+(x_{i+2}+x_{i-2})/4 \ge m(\delta, 2, v_i)\ge t. \end{equation} (5.1)

    for each i\in [0, 4] . This implies that \frac{5}{2}(x_0+x_1+x_2+x_3+x_4)\ge 5t . Thus,

    \begin{equation} \pi_{(2,t)}^*(C_5)\ge 2t. \end{equation} (5.2)

    Note that Eq (3.1) is not always true for C_5 . For example, let \delta = \langle 0, 1, 1, 2, 0 \rangle . Then, we can move one pebble to v_0 by applying three pebbling moves. First, we can move one pebble to v_2 from v_3 , and then move one pebble to v_1 from v_2 . Finally, we can move one pebble to v_0 from v_1 . But, when we use (3.1), we have \left\lfloor \frac{1}{2}(\lfloor \frac{x_3}{2}\rfloor+ x_4)\right\rfloor+x_0+\left \lfloor \frac{1}{2}(x_1+\lfloor \frac{x_2}{2}\rfloor)\right \rfloor = 0 . So, Eq (3.1) should be modified into the following:

    \begin{equation} m(\delta, 2, v_i) = \max\{m_{i,j} | j = 1,2,3\}, \end{equation} (5.3)

    where

    m_{i,1} = \left\lfloor (\lfloor x_{i-2}/2\rfloor+x_{i-1})/2\right\rfloor+x_i+\left \lfloor (x_{i+1}+\lfloor x_{i+2}/2\rfloor)/2\right \rfloor,
    \begin{equation*} m_{i,2} = \begin{cases} m_{i,1},&\qquad \text{if } x_{i-2} \le 1;\\ \left\lfloor\frac {\lfloor (x_{i-2}-2)/2\rfloor+x_{i-1}}{2}\right\rfloor+x_i+\left \lfloor \frac{x_{i+1}+\lfloor (x_{i+2}+1)/2\rfloor}{2}\right \rfloor, & \qquad \text{otherwise,} \end{cases} \end{equation*}

    and

    \begin{equation*} m_{i,2} = \begin{cases} m_{i,1}, &\qquad \text{if } x_{i+2} \le 1;\\ \left\lfloor \frac {\lfloor (x_{i-2}+1)/2\rfloor+x_{i-1}}{2}\right\rfloor+x_i+\left \lfloor \frac{x_{i+1}+\lfloor (x_{i+2}-2)/2\rfloor}{2}\right \rfloor,& \qquad \text{otherwise.} \end{cases} \end{equation*}

    By Corollary 2.1, we have the following.

    Proposition 5.1. \pi_{(2, 10k)}^*(C_5) = 20k for k\ge 1 .

    By Proposition 5.1, we have the following.

    Lemma 5.1. \pi_{(2, 10k+r)}^*(C_5)\le 20k+\pi_{(2, r)}^*(C_5) for k\ge 1 and r\ge 1 .

    Now, we will give a lower bound for \pi_{(2, t)}^*(C_5) when t \mod 10 \neq 0 .

    Lemma 5.2. \pi_{(2, t)}^*(C_5)\ge 2t+1 when t \mod 10 \neq 0 .

    Proof. Let \delta = \langle x_0, x_1, x_2, x_3, x_4 \rangle be an optimal (2, t) -solvable distribution of C_5 . By (5.2), we have \pi_{(2, t)}^*(C_5)\ge 2t . By (5.1), we can write x_i+(x_{i+1}+x_{i-1})/2+(x_{i+2}+x_{i-2})/4 = t+s_i , where s_i is a nonnegative real number for each i\in [0, 4] . If \pi_{(2, t)}^*(C_5) = 2t , then s_i = 0 for all i\in[0, 4] , and this system of linear equations can be represented as a matrix.

    \begin{align*} \left[\begin{matrix} 1&1/2&1/4&1/4&1/2&t\\ 1/2&1&1/2&1/4&1/4&t\\ 1/4&1/2&1&1/2&1/4&t\\ 1/4&1/4&1/2&1&1/2&t\\ 1/2&1/4&1/4&1/2&1&t\\ \end{matrix}\right]. \end{align*}

    After Gaussian elimination, we have

    \begin{align*} \left[\begin{matrix} 1&0&0&0&0&2t/5\\ 0&1&0&0&0&2t/5\\ 0&0&1&0&0&2t/5\\ 0&0&0&1&0&2t/5\\ 0&0&0&0&1&2t/5\\ \end{matrix}\right]. \end{align*}

    Thus, the system of equations has a unique solution x_0 = x_1 = x_2 = x_3 = x_4 = \frac{2t}{5} . If t \mod 5 \neq 0 , then the solution is not integral, hence, there exists at least one i\in [0, 4] such that s_i > 0 . This implies \pi_{(2, t)}^*(C_5)\ge 2t+1 because \pi_{(2, t)}^*(C_5) is an integer. If t \mod 10 = 5 and \pi_{(2, t)}^*(C_5) = 2t , let t = 10k+5 , where k is a nonnegative integer, then \delta = \langle 4k+2, 4k+2, 4k+2, 4k+2, 4k+2 \rangle . It is easy to check that \delta is not (2, t) -solvable. It leads to a contradiction and hence \pi_{(2, t)}^*(C_5)\ge 2t+1 . This completes the proof.

    It is known that \pi^*(C_5) = \lceil \frac{2\times 5}{3}\rceil = 4 , see [2]. So, we have the following.

    Proposition 5.2. \pi_{(2, 1)}^*(C_5) = 4 .

    In 2018, Chen and Shiue [4] showed that \pi^*_{(2, 2)}(C_3) = 4 and \pi^*_{(2, 2)}(C_n) = n for n\ge 4 . So, we have the following.

    Proposition 5.3. \pi_{(2, 2)}^*(C_5) = 5 .

    Lemma 5.3. \pi_{(2, r)}^*(C_5) = 2r+1 for r\in [2, 9] \cup \{11\} .

    Proof. By Proposition 5.3, it is true for r = 2 . By Lemma 5.2, we have \pi_{(2, r)}^*(C_5)\ge 2r+1 for r\in [3, 9] \cup \{11\} . We will prove that the lower bound is also an upper bound by constructing a (2, r) -solvable distribution with 2r+1 pebbles. Let

    \begin{align*} \delta_3& = \langle 1, 2, 1, 2, 1 \rangle,\; \delta_4 = \langle 1, 3, 1, 2, 2 \rangle, \; \delta_5 = \langle 3, 2, 2, 2, 2 \rangle,\\ \delta_6& = \langle 4, 2, 2, 3, 2 \rangle,\; \delta_7 = \langle 3, 3, 3, 3, 3 \rangle,\; \delta_8 = \langle 3, 4, 3, 3, 4 \rangle, \\ \delta_9& = \langle 3, 4, 4, 4, 4 \rangle,\; {\text and} \; \delta_{11} = \langle 5, 4, 5, 5, 4 \rangle. \end{align*}

    Clearly, |\delta_r(C_5)| = 2r+1 . Then, by using (5.3), it is not difficult to verify that \delta_r is a (2, r) -solvable distribution of C_5 for each r\in [3, 9] \cup \{11\} .

    Proposition 5.4. \pi_{(2, 10k+r)}^*(C_5) = 20k+2r+1 for k\ge 0 and r\in [2, 9] \cup \{11\} .

    Proof. By Lemma 5.3, it is true for k = 0 . By combining Proposition 5.1 and Lemma 5.3, we have \pi_{(2, 10k+r)}^*(C_5)\le 20k+\pi_{(2, r)}^*(C_5) = 20k+2r+1 for k\ge 1 . By Lemma 5.2, we can see that the upper bound is also a lower bound, hence, we have the proof.

    By combining Propositions 5.1, 5.2, and 5.4, we have the main result of this section.

    Theorem 5.1. Let t be a positive integer. Then,

    \begin{equation*} \pi_{(2,t)}^*(C_5) = \begin{cases} 4, &\qquad \mathit{\text{ if }} t = 1;\\ 2t, & \qquad \mathit{\text{ if }} t \mod 10 = 0;\\ 2t+1, &\qquad \mathit{\text{ if }} t\ge 2 \mathit{\text{ and }}t\mod 10 \neq 0. \end{cases} \end{equation*}

    In 2018, the distance-restricted pebbling was first proposed by Chen and Shiue [4], and they showed that \pi^*_{(2, 2)}(C_3) = 4 and \pi^*_{(2, 2)}(C_n) = n for n\ge 4 . In 2020, Shiue [13] studied distance 1-restricted pebbling in cycles, and he determined the exact value of \pi^*_{(1, t)}(C_n) for all t\ge 1 and n\ge 3 ; see Theorem 2.1. For t = 1 , \pi^*_{(1, 1)}(C_n) = \pi^*(C_n) = \left\lceil \frac{2n}{3} \right\rceil ; see [2]. This implies that \pi^*_{(d, 1)}(C_n) = \left\lceil \frac {2n}{3} \right\rceil for all d\ge 1 . Thus, we have \pi^*_{(2, 1)}(C_n) = \left\lceil \frac{2n}{3} \right\rceil .

    " (d, t) -pebbling" can be seen as a generalization of optimal pebbling. In addition to this, it is our belief that (d, t) -pebbling is more applicable than other versions of pebbling. For example, in a transportation and resource allocation system, the resource must be delivered to a target in a set amount of time, thus, in practicality, we need to restrict the distance to the target. Ideally, in such a resource allocation scheme, the storehouses for the resources should have a capacity restriction since the space for the storehouse is limited. Studies on capacity-restricted optimal pebbling can be found in [3,10]. We discussed the optimal capacity and distance-restricted t -fold pebbling, also known as the optimal (c, d, t) -pebbling number, in [15]. In some sense, \pi^*_{(d, t)}(G) can also be viewed as a distance-based parameter, which is a topic that has been studied before, for example, in [1].

    In this article, we give a further study into " (d, t) -pebbling". Our result makes progress towards understanding (d, t) -pebbling for cycles. Specifically, we study (2, t) -pebbling as a continuation from the study of (1, t) -pebbling done in [13]. We were able to show that the optimal (2, t) -pebbling number of cycles can be determined by considering only a finite number of cases, namely, 1\le t \le 12 . We were also able to completely determine the optimal (2, t) -pebbling number for C_4 and C_5 . For cycles of length greater than 5, our result relies on repeated application of the tools developed in Facts 1–10. For cycles of length not greater than 5, the key to our results is obtained by modifying Eq (3.1). The process of obtaining a lower bound for \pi^*_{(2, t)}(C_5) involves the solution of a system of linear equations whose matrix is positive semi-definite. For arbitrarily large distance d , the matrix can be very large. Thus, in order to obtain a lower bound for \pi^*_{(d, t)}(C_n) when d > 2 , one may consider the use of the conjugate gradient method; see [6].

    Finally, we will give a summary of progress towards the determination of \pi^*_{(2, t)}(C_n) for all n\ge 3 and t\ge 1 . Corollary 3.1 implies that the problem of determining the exact value of \pi^*_{(2, t)}(C_n) for n\ge 6 and t\ge 1 can be reduced to the problem of determining the exact value of \pi^*_{(2, r)}(C_n) for n\ge 6 and r\in[1, 12] . By the discussion above, we know the exact value of \pi^*_{(2, t)}(C_n) for n\ge 3 and t = 1, 2 . Furthermore, note that the diameter of C_3 is equal to one. This implies that \pi^*_{(2, t)}(C_3) = \pi^*_{(1, t)}(C_3) for t\ge 1 , see Theorem 2.1. Theorems 4.1 and 5.1 give the exact value of \pi^*_{(2, t)}(C_n) for n = 4, 5 and t\ge 1 . So, to completely determine the exact value of \pi^*_{(2, t)}(C_n) for n\ge 3 and t\ge 1 , it is left to solve the following problem.

    Problem 2. Determine the exact value of \pi^*_{(2, t)}(C_n) for n\ge 6 and 3 \le t \le 12 .

    Recently, we have shown that \pi^*_{(2, 3)}(C_n) = \lceil \frac{4n}{3}\rceil for n\ge 4 , see [15]. By combining Corollary 3.1, we have \pi^*_{(2, 10k+3)}(C_n) = \lceil \frac{4n}{3}\rceil+4kn for n\ge 6 and k\ge 0 . For example, by using the two facts that \langle 4, 0, 0, 4, 0, 0\rangle is an optimal (2, 3) -solvable distribution of C_6 (see [15]) and that \langle 4, 4, 4, 4, 4, 4\rangle is an optimal (2, 10) -solvable distribution of C_6 (see [13]), we can conclude that \langle 8, 4, 4, 8, 4, 4\rangle is an optimal (2, 13) -solvable distribution of C_6 .

    Chin-Lin Shiue: Conceptualization, Supervision, Writing–original draft; Tzu-Hsien Kwong: Conceptualization, Validation, Writing–review and editing. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This research was supported in part by the Ministry of Science and Technology of Taiwan under Grant MOST 111-2115-M-033-003.

    The authors declare that there are no conflicts of interest in this paper.



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