@article{Liu2026, 
author = {Siyue Liu and Gang Chen},
title = {Two disjoint cycles with prescribed lengths and arcs in digraphs},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {3},
pages = {6560-6568},
keywords = {directed graph, disjoint dicycles, cycle coverings},
url = {https://www.sciopen.com/article/10.3934/math.2026271},
doi = {10.3934/math.2026271},
abstract = {Let    D  =  (  V  ,  A  ) be an    n-vertex digraph with    n  ≥  6. For each vertex    v  ∈  V, let    a  (  v  ) be the degree of    v in    D. Assume that for every pair of distinct vertices    x  ,  y  ∈  V, the sum of their degrees satisfies    a  (  x  )  +  a  (  y  )  ≥  3  n  +  1. For any two independent arcs        f    1    ,      f    2    ∈  A  (  D  ) and any integer partition    n  =      n    1    +      n    2   where        n    1    ≥  3 and        n    2    ≥  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    ∈  E  (      C    1    ), and        f    2    ∈  E  (      C    2    ). Moreover, the condition is sharp.}
}