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Gallai's path decomposition conjecture for block graphs

  • Let G be a graph of order n. A path decomposition P of G is a collection of edge-disjoint paths that covers all the edges of G. Let p(G) denote the minimum number of paths needed in a path decomposition of G. Gallai conjectured that if G is connected, then p(G)n2. In this paper, we prove that the above conjecture holds for all block graphs.

    Citation: Xiaohong Chen, Baoyindureng Wu. Gallai's path decomposition conjecture for block graphs[J]. AIMS Mathematics, 2025, 10(1): 1438-1447. doi: 10.3934/math.2025066

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  • Let G be a graph of order n. A path decomposition P of G is a collection of edge-disjoint paths that covers all the edges of G. Let p(G) denote the minimum number of paths needed in a path decomposition of G. Gallai conjectured that if G is connected, then p(G)n2. In this paper, we prove that the above conjecture holds for all block graphs.



    1. Introduction

    G. Caginalp proposed in [3] and [4] two phase-field system, namely,

    utΔu+f(u)=T, (1.1)
    TtΔT=ut, (1.2)

    called nonconserved system, and

    ut+Δ2uΔf(u)=ΔT, (1.3)
    TtΔT=ut, (1.4)

    called concerved system (in the sense that, when endowed with Neumann boundary conditions, the spacial average of u is conserved). In this context, u is the order parameter, T is the relative temperature (defined as T=˜TTE, where ˜T is the absolute temperature and TE is the equilibrium melting temperature) and f is the derivative of a double-well potential F (a typical choice is F(s)=14(s21)2, hence the usual cubic nonlinear term f(s)=s3s). Furthermore, we have set all physical parameters equal to one. These systems have been introduced to model phase transition phenomena, such as melting-solidication phenomena, and have been much studied from a mathematical point of view. We refer the reader to, e.g., [3,4,5,8,9,10,12,13,14,15,16,18,19,21,22,23,25].

    Both systems are based on the (total Ginzburg-Landau) free energy

    ΨGL=Ω(12|u|2+F(u)uT12T2)dx, (1.5)

    where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R3, with boundary Γ), and the enthalpy

    H=u+T. (1.6)

    As far as the evolution equations for the order parameter are concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one)

    uu=DΨGLDu, (1.7)

    for the nonconserved model, and

    uu=ΔDΨGLDu, (1.8)

    for the conserved one, where DDu denotes a variational derivative with respect to u, which yields (1.1) and (1.3), respectively. Then, we have the energy equation

    Ht=divq, (1.9)

    where q is the heat flux. Assuming finally the usual Fourier law for heat conduction,

    q=T, (1.10)

    we obtain (1.2).

    In (1.5), the term |u|2 models short-ranged interactions. It is however interesting to note that such a term is obtained by truncation of higher-order ones; it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions [11].

    G. Caginalp and Esenturk recently proposed in [6] (see also [20]) higher-order phase-field models in order to account for anisotropic interfaces (see also [7] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these autors proposed the following modified (total) free energy

    ΨHOGL=Ω(12ki=1|β|=iaβ|Dβu|2+F(u)uT12T2)dx,kN, (1.11)

    where, for β=(k1,k2,k3)(N{0})3,

    |β|=k1+k2+k3

    and, for β(0,0,0),

    Dβ=|β|xk11xk22xk33

    (we agree that D(0,0,0)v=v).

    A. Miranville studied in [17] the corresponding nonconserved higher-order phase-field system.

    As far as the conserved case is concerned, the above generalized free energy yields, procceding as above, the following evolution equation for the order parameter u:

    utΔki=1(1)i|β|=iaβD2βuΔf(u)=Δ(αtΔαt), (1.12)

    In particular, for k = 1 (anisotropic conserved Caginalp phase-field), we have an equation of the form

    ut+Δ3i=1ai2ux2iΔf(u)=Δ(αtΔαt)

    and, for k = 2 (fourth-order anisotropic conserved Caginalp phase-field system), we have an equation of the form

    utΔ3i,j=1aij4ux2ix2j+Δ3i=1bi2ux2iΔf(u)=Δ(αtΔαt).

    L. Cherfils A. Miranville and S. Peng have studied in [8] the corresponding higher-order isotropic equation (without the coupling with the temperature), namely, the equation

    utΔP(Δ)uΔf(u)=0,

    where

    P(s)=ki=1aisi,ak>0,k1,

    endowed with the Dirichlet/Navier boundary conditions

    u=Δu=...=Δku=0onΓ.

    Our aim in this paper is to study the model consisting of the higher-order anisotropic equation (1.12) and the temperature equation

    2αt2Δ2αt2ΔαtΔα=ut. (1.13)

    In particular, we obtain the existence and uniqueness of solutions.


    2. Setting of the problem

    We consider the following initial and boundary value problem, for kN, k2 (the case k = 1 can be treated as in the original conserved system; see, e.g., [23]):

    utΔki=1(1)i|β|=iaβD2βuΔf(u)=Δ(αtΔαt), (2.1)
    2αt2Δ2αt2ΔαtΔα=ut, (2.2)
    Dβu=α=0onΓ,|β|k, (2.3)
    u|t=0=u0,α|t=0=α0,αt|t=0=α1. (2.4)

    We assume that

    aβ>0,|β|=k, (2.5)

    and we introduce the elliptic operator Ak defined by

    Akv,wHk(Ω),Hk0(Ω)=|β|=kaβ((Dβv,Dβw)), (2.6)

    where Hk(Ω) is the topological dual of Hk0(Ω). Furthermore, ((., .)) denotes the usual L2-scalar product, with associated norm .. More generally, we denote by .X the norm on the Banach space X; we also set .1=(Δ)12., where (Δ)1 denotes the inverse minus Laplace operator associated with Dirichlet boudary conditions. We can note that

    (v,w)Hk0(Ω)2|β|=kaβ((Dβv,Dβw))

    is bilinear, symmetric, continuous and coercive, so that

    Ak:Hk0(Ω)Hk(Ω)

    is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order 2k (see [1] and [2]) that Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

    D(Ak)=H2k(Ω)Hk0(Ω),

    where, for vD(Ak),

    Akv=(1)k|β|=kaβD2βv.

    We further note that D(A12k)=Hk0(Ω) and, for (v,w)D(A12k)2,

    ((A12kv,A12kw))=|β|=kaβ((Dβv,Dβw)).

    We finally note that (see, e.g., [24]) Ak. (resp., A12k.) is equivalent to the usual H2k-norm (resp., Hk-norm) on D(Ak) (resp., D(A12k)).

    Similarly, we can define the linear operator ¯Ak=ΔAk

    ˉAk:Hk+10(Ω)Hk1(Ω)

    which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

    D(ˉAk)=H2k+2(Ω)Hk+10(Ω),

    where, for vD(ˉAk),

    ˉAkv=(1)k+1Δ|β|=kaβD2βv.

    Furthermore, D(ˉA12k)=Hk+10(Ω) and, for (v,w)D(ˉA12k),

    ((ˉA12kv,ˉA12kw))=|β|=kaβ((Dβv,Dβw)).

    Besides ˉAk. (resp., ˉA12k.) is equivalent to the usual H2k+2-norm (resp., Hk+1-norm) on D(ˉAk) (resp., D(ˉA12k)).

    We finally consider the operator ˜Ak=(Δ)1Ak, where

    ˜Ak:Hk10(Ω)Hk+1(Ω);

    note that, as Δ and Ak commute, then the same holds for (Δ)1 and Ak, so that ˜Ak=Ak(Δ)1.

    We have the (see [17])

    Lemme 2.1. The operator ˜Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

    D(˜Ak)=H2k2(Ω)Hk10(Ω),

    where, for vD(˜Ak)

    ˜Akv=(1)k|β|=kaβD2β(Δ)1v.

    Furthermore, D(˜A12k)=Hk10(Ω) and, for (v,w)D(˜A12k),

    ((˜A12kv,˜A12kw))=|β|=kaβ((Dβ(Δ)12v,Dβ(Δ)12w)).

    Besides ˜Ak. (resp., ˜A12k.) is equivalent to the usual H2k2-norm (resp., Hk1-norm) on D(˜Ak) (resp., D(˜A12k)).

    Proof. We first note that ˜Ak clearly is linear and unbounded. Then, since (Δ)1 and Ak commute, it easily follows that ˜Ak is selfadjoint.

    Next, the domain of ˜Ak is defined by

    D(˜Ak)={vHk10(Ω),˜AkvL2(Ω)}.

    Noting that ˜Akv=f,fL2(Ω),vD(˜Ak), is equivalent to Akv=Δf, where ΔfH2(Ω), it follows from the elliptic regularity results of [1] and [2] that vH2k2(Ω), so that D(˜Ak)=H2k2(Ω)Hk10(Ω).

    Noting then that ˜A1k maps L2(Ω) onto H2k2(Ω) and recalling that k2, we deduce that ˜Ak has compact inverse.

    We now note that, considering the spectral properties of Δ and Ak (see, e.g., [24]) and recalling that these two operators commute, Δ and Ak have a spectral basis formed of common eigenvectors. This yields that, s1,s2R, (Δ)s1 and As2k commute.

    Having this, we see that ˜A12k=(Δ)12A12k, so that D(˜A12k)=Hk10(Ω), and for (v,w)D(˜A12k)2,

    ((˜A12kv,˜A12kw))=|β|=kaβ((Dβ(Δ)12v,Dβ(Δ)12w)).

    Finally, as far as the equivalences of norms are concerned, we can note that, for instance, the norm ˜A12k. is equivalent to the norm (Δ)12.Hk(Ω) and, thus, to the norm (Δ)k12..

    Having this, we rewrite (2.1) as

    utΔAkuΔBkuΔf(u)=Δ(αtΔαt), (2.7)

    where

    Bkv=k1i=1(1)i|β|=iaβD2βv.

    As far as the nonlinear term f is concerned, we assume that

    fC2(R),f(0)=0, (2.8)
    fc0,c00, (2.9)
    f(s)sc1F(s)c2c3,c1>0,c2,c30,sR, (2.10)
    F(s)c4s4c5,c4>0,c50,sR, (2.11)

    where F(s)=s0f(τ)dτ. In particular, the usual cubic nonlinear term f(s)=s3s satisfies these assumptions.

    Throughout the paper, the same letters c, c' and c" denote (generally positive) constants which may vary from line to line. Similary, the same letter Q denotes (positive) monotone increasing (with respect to each argument) and continuous functions which may vary from line to line.


    3. A priori estimates

    We multiply (2.7) by (Δ)1ut and (2.2) by αtΔαt, sum the two resulting equalities and integrate over Ω and by parts. This gives

    ddt(A12ku2+B12k[u]+2ΩF(u)dx+α2+Δα2+αtΔαt2)+2ut21+2αt2+2Δαt2=0 (3.1)

    (note indeed that αt2+2αt2+Δαt2=αtΔαt2), where

    B12k[u]=k1i=1|β|=iaβDβu2 (3.2)

    (note that B12k[u] is not necessarily nonnegative). We can note that, owing to the interpolation inequality

    B12k[u]=k1i=1|β|=iaβDβu2 (3.3)
    (Δ)i2vc(i)(Δ)m2vimv1im,

    there holds

    vHm(Ω),i{1,...,m1},mN,m2, (3.4)

    This yields, employing (2.11),

    |B12k[u]|12A12ku2+cu2.

    whence

    A12ku2+B12k[u]+2ΩF(u)dx12A12ku2+ΩF(u)dx+cu4L4(Ω)cu2c", (3.5)

    nothing that, owing to Young's inequality,

    A12ku2+B12k[u]+2ΩF(u)dxc(u2Hk(Ω)+ΩF(u)dx)c,c>0, (3.6)

    We then multiply (2.7) by (Δ)1u and have, owing to (2.10) and the interpolation inequality (3.3),

    u2ϵu4L4(Ω)+c(ϵ),ϵ>0.

    hence, proceeding as above and employing, in particular, (2.11)

    ddtu21+c(u2Hk(Ω)+ΩF(u)dx)c(u2+αt2+Δαt2)+c", (3.7)

    Summing (3.1) and δ1 times (3.7), where δ1>0 is small enough, we obtain a differential inegality of the form

    ddtu21+c(u2Hk(Ω)+ΩF(u)dx)c(αt2+Δαt2)+c,c>0. (3.8)

    where

    ddtE1+c(u2Hk(Ω)+ΩF(u)dx+ut21+αt2H2(Ω))c,c>0,

    satisfies, owing to (3.5)

    E1=A12ku2+B12k[u]+2ΩF(u)dx+α2+Δα2+αtΔαt2+δ1u21 (3.9)

    Multiplying (2.2) by Δα, we then obtain

    E1c(u2Hk(Ω)+ΩF(u)dx+α2H2(Ω)+αt2H2(Ω))c,c>0.

    which yields, employing the interpolation inequality

    ddt(Δα22((αt,Δα))+2((Δαt,Δα)))+Δα2ut2+αt2+Δαt2, (3.10)

    the differential inequality, with 0<ϵ<<1 is small enough

    v2cv1vH1(Ω),vH10(Ω), (3.11)

    We now differentiate (2.7) with respect to time to find, owing to (2.2),

    ddt(Δα22((αt,Δα))+2((Δαt,Δα)))+cα2H2(Ω)c(ut21+ϵut2H1(Ω)+αt2H2(Ω)),c>0. (3.12)

    together with the boundary condition

    tutΔAkutΔBkutΔ(f(u)ut)=Δ(Δαt+Δαut), (3.13)

    We multiply (3.11) by (Δ)1ut and obtain, owing to (2.9) and the interpolation inequality (3.3),

    Dβut=0onΓ,|β|k.

    hence, owing to (3.10), the differential inequality

    ddtut21+cut2Hk(Ω)c(ut2+Δα2+Δαt2),c>0, (3.14)

    Summing finally (3.8), δ2 times (3.11) and δ3 times (3.14), where δ2,δ3>0 are small enough, we find a differential inequality of the form

    ddtut21+cut2Hk(Ω)c(ut21+α2H2(Ω)+αt2H2(Ω)),c>0. (3.15)

    where

    dE2dt+c(E2+ut2Hk(Ω))c,c>0,

    Owing to the continuous embedding H2k+1(Ω)C(ˉΩ), we deduce that

    E2=E1+δ2(Δα22((αt,Δα))+2((Δαt,Δα)))+δ3ut21.

    and since

    |ΩF(u0)dx|Q(u0H2k+1(Ω))

    we see that (Δ)12ut(0)L2(Ω) and

    (Δ)12ut(0)=(Δ)12Aku0(Δ)12Bku0(Δ)12f(u0)+(Δ)12(α1Δα1), (3.16)

    Furthermore E2 satisfies

    ut(0)1Q(u0H2k+1(Ω),α1H3(Ω)). (3.17)

    It thus follows from (3.15), (3.16), (3.17) and Growall's lemma that

    E2c(u2Hk(Ω)+ut21+ΩF(u)dx+α2H2(Ω)+αt2H2(Ω))c,c>0. (3.18)

    and

    u(t)2Hk(Ω)+ut(t)21+α(t)2H2(Ω)+αt(t)2H2(Ω)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c>0,t0, (3.19)

    r>0 given.

    Multiplying next (2.7) by ˜Aku, we find, owing to the interpolation inequality (3.3),

    t+rtut2Hk(Ω)dsectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c(r),c>0,t0,

    hence, since f and F are continuous and owing to (3.18),

    ddt˜A12ku2+cu2H2k(Ω)c(u2+f(u)2+αt2+Δαt2),c>0, (3.20)

    Summing (3.15) and (3.22), we have a differential inequality of the form

    ddt˜A12ku2+cu2H2k(Ω)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c",c,c>0,t0. (3.21)

    where

    dE3dt+c(E3+u2H2k(Ω)+ut2Hk(Ω))ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c",c,c>0,t0,

    satisfies

    E3=E2+˜A12ku2 (3.22)

    In particular, it follows from (3.21)-(3.22) that

    E3c(u2Hk(Ω)+ut21+ΩF(u)dx+α2H2(Ω)+αt2H2(Ω))c,c>0. (3.23)

    r>0 given.

    We now multiply (2.7) by u and obtain, employing (2.9) and the interpolation inequality (3.3)

    t+rtu2H2k(Ω)dsectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c(r),c>0,t0,

    whence, proceeding as above,

    ddtu2+cu2Hk+1(Ω)c(u2H1(Ω)+αt2+Δαt2),c>0, (3.24)

    We also multiply (2.7) by ut and find

    ddtu2+cu2Hk+1(Ω)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c,c>0.

    where

    ddt(ˉA12ku2+ˉB12k[u])+cut2cΔf(u)22((Δut,αtΔαt)),

    Since f is of class C2, it follows from the continuous embedding H2(Ω)C(ˉΩ) that

    ˉB12k[u]=k1i=1|β|=iaβDβu2.

    hence, owing to (3.18),

    Δf(u)2Q(uH2(Ω)), (3.25)

    Multiply next (2.2) by Δ(αtΔαt), we have

    ddt(ˉA12ku2+ˉB12k[u])+cut2ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))2((Δut,αtΔαt))+c,c,c>0. (3.26)

    (note indeed that αt2+2Δαt2+Δαt2=αtΔαt2).

    Summing (3.25) and (3.26), we obtain

    ddt(Δα2+Δα2+αtΔαt2)+c(Δαt2+Δαt2)2((Δut,αtΔαt)),c>0 (3.27)

    Summing finally (3.21), (3.24) and (3.27), we find a differential inegality of the form

    ddt(ˉA12ku2+ˉB12k[u]+Δα2+Δα2+αtΔαt2)+c(ut2+Δαt2+Δαt2)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c,c>0. (3.28)

    where

    dE4dt+c(E3+u2Hk+1(Ω)+u2H2k(Ω)+ut2+ut2Hk(Ω)+αt2H3(Ω))ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c,c>0,t0

    satisfies, owing to (2.11) and the interpolation inegality (3.3)

    E4=E3+u2+ˉA12ku2+ˉB12k[u]+Δα2+Δα2+αtΔαt2 (3.29)

    In particular, it follows from (3.28)-(3.29) that

    E4c(u2Hk+1(Ω)+ut21+ΩF(u)dx+α2H3(Ω)+αt2H3(Ω))c,c>0. (3.30)

    and

    u(t)Hk+1(Ω)+α(t)H3(Ω)+αt(t)H3(Ω)ectQ(u0H2k+1(Ω),α0H3(Ω),α1H3(Ω))+c,c>0,t0, (3.31)

    r given.

    We finally rewrite (2.7) as an elliptic equation, for t > 0 fixed,

    t+rt(ut2+αt2H3(Ω))dsectQ(u0H2k+1(Ω),α0H3(Ω),α1H3(Ω))+c(r),c>0,t0, (3.32)

    Multiplying (3.32) by Aku, we obtain, owing to the interpolation inequality (3.3),

    Aku=(Δ)1utBkuf(u)+αtΔαt,Dβu=0onΓ,|β|k1.

    hence, since f is continuous and owing to (3.18)

    Aku2c(u2+f(u)2+ut21+αt2+Δαt2), (3.33)

    4. Existence and uniqueness of solutions

    We first have the following theorem.

    Theorem 4.1. (i) We assume that (u0,α0,α1)Hk0(Ω)×(H2(Ω)H10(Ω))×(H2(Ω)H10(Ω)), with ΩF(u0)dx<+. Then, (2.1)(2.4) possesses at last one solution (u,α,αt) such that, T>0, u(0)=u0, α(0)=α0, αt(0)=α1,

    u(t)2H2k(Ω)cectQ(u0H2k+1(Ω),α0H3(Ω),α1H3(Ω))+c,c>0t0.
    uL(R+;Hk0(Ω))L2(0,T;H2k(Ω)Hk0(Ω)),
    utL(R+;H1(Ω))L2(0,T;Hk0(Ω)),

    and

    α,αtL(R+;H2(Ω)H10(Ω))
    ddt((Δ)1u,v))+ki=1|β|=iai((Dβu,Dβv))+((f(u),v))=ddt(((u,v))+((u,v))),vCc(Ω),

    in the sense of distributions.

    (ii) If we futher assume that (u0,α0,α1)(Hk+1(Ω)Hk0(Ω))×(H3(Ω)H10(Ω))×(H3(Ω)H10(Ω)), then, T>0,

    ddt(((αt,w))+((αt,w))+((α,w)))+((α,w))=ddt((u,w)),wCc(Ω),
    uL(R+;Hk+1(Ω)Hk0(Ω))L2(R+;Hk+1(Ω)Hk0(Ω))
    utL2(R+;L2(Ω)),

    and

    αL(R+;H3(Ω)H10(Ω))

    The proofs of existence and regularity in (i) and (ii) follow from the a priori estimates derived in the previous section and, e.g., a standard Galerkin scheme.

    We then have the following theorem.

    Theorem 4.2. The system (1.1)-(1.4) possesses a unique solution with the above regularity.

    proof. Let (u(1),α(1),α(1)t) and (u(2),α(2),α(2)t) be two solutions to (2.1)-(2.3) with initial data (u(1)0,α(1)0,α(1)1) and (u(2)0,α(2)0,α(2)1), respectively. We set

    αtL(R+;H3(Ω)H10(Ω))L2(0,T;H3(Ω)H10(Ω))

    and

    (u,α,αt)=(u(1),α(1),α(1)t)(u(2),α(2),α(2)t)

    Then, (u,α) satisfies

    (u0,α0,α1)=(u(1)0,α(1)0,α(1)1)(u(2)0,α(2)0,α(2)1). (4.1)
    utΔAkuΔBkuΔ(f(u(1))f(u(2)))=Δ(αtΔαt), (4.2)
    2αt2Δ2αt2ΔαtΔα=ut, (4.3)
    Dβu=α=0 on Γ,|β|k, (4.4)

    Multiplying (4.1) by (Δ)1u and integrating over Ω, we obtain

    u|t=0=u0,α|t=0=α0,αt|t=0=α1.

    We note that

    ddtu21+cu2Hk(Ω)c(u2+αtΔαt2)2((f(u(1))f(u(2),u)).

    with l defined as

    f(u(1))f(u(2))=l(t)u,

    Owing to (2.9), we have

    l(t)=10f(su(1)(t)+(1s)u(2)(t))ds.

    and we obtain owing to the intepolation inequalities (3.3) and (3.10),

    2((f(u(1))f(u(2),u))2c0u2                                      cu2 (4.5)

    Next, multiplying (4.2) by (Δ)1(u+αtΔαt), we find

    ddtu21+cu2Hk(Ω)c(u21+αtΔαt2),c>0. (4.6)

    Summing then δ4 times (4.5) and (4.6), where δ4>0 is small enough, we have, employing once more the interpolation inequality (3.10), a differential inequality of the form

    ddt(α2+α2+u+αtΔαt21)+c(αt2+αt2H1(Ω))c(u2+α2). (4.7)

    where

    dE5dtcE5,

    satisfies

    E5=δ4u21+α2+α2+u+αtΔαt21 (4.8)

    It follows from (4.7)-(4.8) and Gronwall's lemma that

    E5c(u21+α2H1(Ω)+αtΔαt2),c>0. (4.9)

    hence the uniquess, as well as the continuous dependence with respect to the initial data in H1×H1×H1-norm.


    Conflict of Interest

    All authors declare no conflicts of interest in this paper.




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