Disjoint Cycles of Different Lengths in Graphs and Digraphs

In this paper, we study the question of finding a set of k vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every k > 1, every graph with minimum degree at least k 2+5k−2 2 has k vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider other cases such as when ∗This paper is dedicated to the memory of Nicolas Lichiardopol. †Supported by ERC Advanced Grant GRACOL, project no. 320812. ‡Supported by an FQRNT postdoctoral research grant and CIMI research fellowship. §Supported by NSFC (11601429, 11671320) and the Natural Science Foundation of Shaanxi Province (2016JQ1002, 2014JK1353). the electronic journal of combinatorics 24(4) (2017), #P4.37 1 the graph is triangle-free, or the k cycles are required to have different lengths modulo some value r. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have k vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for some regular digraphs and digraphs of small order.