Let $ D = (V, A) $ be an $ n $-vertex digraph with $ n \geq 6 $. For each vertex $ v \in V $, let $ a(v) $ be the degree of $ v $ in $ D $. Assume that for every pair of distinct vertices $ x, y \in V $, the sum of their degrees satisfies $ a(x) + a(y) \geq 3n + 1 $. For any two independent arcs $ f_1, f_2 \in A(D) $ and any integer partition $ n = n_1 + n_2 $ where $ n_1 \geq 3 $ and $ n_2 \geq 3 $, $ D $ contains two mutually vertex-disjoint dicycles $ C_1 $ and $ C_2 $ such that $ |V(C_1)| = n_1 $, $ |V(C_2)| = n_2 $, $ f_1 \in E(C_1) $, and $ f_2 \in E(C_2) $. Moreover, the condition is sharp.
Citation: Siyue Liu, Gang Chen. Two disjoint cycles with prescribed lengths and arcs in digraphs[J]. AIMS Mathematics, 2026, 11(3): 6560-6568. doi: 10.3934/math.2026271
Let $ D = (V, A) $ be an $ n $-vertex digraph with $ n \geq 6 $. For each vertex $ v \in V $, let $ a(v) $ be the degree of $ v $ in $ D $. Assume that for every pair of distinct vertices $ x, y \in V $, the sum of their degrees satisfies $ a(x) + a(y) \geq 3n + 1 $. For any two independent arcs $ f_1, f_2 \in A(D) $ and any integer partition $ n = n_1 + n_2 $ where $ n_1 \geq 3 $ and $ n_2 \geq 3 $, $ D $ contains two mutually vertex-disjoint dicycles $ C_1 $ and $ C_2 $ such that $ |V(C_1)| = n_1 $, $ |V(C_2)| = n_2 $, $ f_1 \in E(C_1) $, and $ f_2 \in E(C_2) $. Moreover, the condition is sharp.
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Siyue Liu, Gang Chen. Two disjoint cycles with prescribed lengths and arcs in digraphs[J]. AIMS Mathematics, 2026, 11(3): 6560-6568. doi: 10.3934/math.2026271
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