The dichromatic number χ⃗false(Dfalse) of a digraph D is the least number k such that the vertex set of D can be partitioned into k parts each of which induces an acyclic subdigraph. Introduced by Neumann‐Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this article, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as Ω(prefixlog2n). We finally give a Brooks‐type upper bound on the list dichromatic number of digon‐free digraphs.
A graph G is said to be ISK4-free if it does not contain any subdivision of K 4 as an induced subgraph. In this paper, we propose new upper bounds for chromatic number of ISK4-free graphs and {ISK4, triangle}-free graphs.
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,
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