Disjoint Cycles of Different Lengths in Graphs and Digraphs

Abstract

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. 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 stronger situations, and exhibit degree bounds (some of which are best possible) when e.g. the graph is triangle-free, or the k cycles are requested to have different lengths congruent to some values modulo some 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 regular digraphs and digraphs of small order. 2 for every k (Theorem 2.12). Several more constrained situations are then considered, e.g. when the graph is trianglefree or the vertex-disjoint cycles are requested to be more than just of different lengths; in these situations as well, we exhibit bounds (most of which are tight) on the least minimum degree required to guarantee the existence of the k desired vertex-disjoint cycles. We also consider the opposite direction, and conjecture that for every D ≥ 3, every graph G verifying k + 1 ≤ δ(G) ≤ ∆(G) ≤ D and of large enough order has k vertex-disjoint cycles of different lengths (see Conjecture 2.18). To support this conjecture, we prove it for k = 2 (Theorem 2.19). This in particular yields that every cubic graph of order more than 14 has two vertex-disjoint cycles of different lengths, which is tight (see Theorem 2.20).We then consider, in Section 3, the same question for digraphs: What minimum outdegree is required for a digraph to have at least k vertex-disjoint directed cycles of different lengths? The existence of such a minimum out-degree was conjectured by Lichiardopol in [9], who verified it for k = 2. We here give further support to Lichiardopol's Conjecture by showing it to hold for tournaments (see Corollary 3.6), and, using the probabilistic method, for regular digraphs (Theorem 3.10) and digraphs of small order (Theorem 3.11). Disjoint cycles of different lengths in undirected graphsIn this section, we consider the existence of disjoint cycles of different lengths in graphs. We start off, in Section 2.1, by recalling a few results and introducing new results and concepts of independent interest. We then prove our main results in Section 2.2. PreliminariesLet G be a graph and X a subset of V (G). We use G[X] to denote the subgraph of G induced by X, and G − X to denote the subgraph of G induced by V (G)\X. For two disjoint subsets X,

A decomposition for digraphs with minimum outdegree 3 having no vertex disjoint cycles of different lengths

“…By repeating these arguments to the vertices v 2 , u, w 3 and then to the vertices w 3 , v 1 , v 2 , we can see that v 1 v 2 ∈ A and uw 3 ∈ A. Now, it is not difficult to see that D is isomorphic to D 2 4 in this case. Case 2.…”

“…C 0 ) = ∅, say u 2 = u 0 , then v 2 = v 0 because D has no vertex which lies on two different cycles of length 2 in this case. Then it is not difficult to see that D is 8 N.D. Tan isomorphic to D 2 4 , which is impossible in this case. Thus, V (C 2 ) ∩ V (C 0 ) = ∅.…”

“…If V (C 3 ) ∩ V (C 0 ) = ∅, say u 3 = u 0 , then since D has no vertex which lies on two different cycles of length 2 in this case we must have v 3 = v 0 , i.e., V (C 3 ) = V (C 0 ). Then, it is not difficult to see that in this situaton D is isomorphic to D 2 6 . Otherwise, V (C 3 ) ∩ V (C 0 ) = ∅ and we can repeat similar arguments as above for C 3 and so on.…”

“…The digraph D 2 4 is the complete digraph on 4 vertices. The digraph D 2 8 is illustrated in Figure 1.…”

“…It has been proved in [7] that for any integer n ≥ 2, the digraph D 2 2n is a 3-regular digraph with girth 2 having no vertex disjoint cycles of different lengths. For more results about digraphs having no vertex disjoint cycles of different lengths, the reader can see the papers [2,8] and [9].…”