A classical theorem of Gallai states that in every graph that is critical for k-colorings, the vertices of degree k − 1 induce a tree-like graph whose blocks are either complete graphs or cycles of odd length. We provide a generalization to colorings and list colorings of digraphs, where some new phenomena arise. In particular, the problem of list coloring digraphs with the lists at each vertex v having min{d + (v), d − (v)} colors turns out to be NP-hard.

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.

It is known (Bollobás [4]; Kostochka and Mazurova [12]) that there exist graphs of maximum degree ∆ and of arbitrarily large girth whose chromatic number is at least c∆/ log ∆. We show an analogous result for digraphs where the chromatic number of a digraph D is defined as the minimum integer k so that V (D) can be partitioned into k acyclic sets, and the girth is the length of the shortest cycle in the corresponding undirected graph. It is also shown, in the same vein as an old result of Erdős [5], that there are digraphs with arbitrarily large chromatic number where every large subset of vertices is 2-colorable.

Brooks' Theorem states that a connected graph G of maximum degree ∆ has chromatic number at most ∆, unless G is an odd cycle or a complete graph. A result of Johansson [6] shows that if G is triangle-free, then the chromatic number drops to O(∆/ log ∆). In this paper, we derive a weak analog for the chromatic number of digraphs. We show that every (loopless) digraph D without directed cycles of length two has chromatic number χ(D) ≤ (1 − e −13 )∆, where∆ is the maximum geometric mean of the out-degree and in-degree of a vertex in D, when∆ is sufficiently large. As a corollary it is proved that there exists an absolute constant α < 1 such that χ(D) ≤ α(∆ + 1) for everỹ ∆ > 2.Keywords: Digraph coloring, dichromatic number, Brooks theorem, digon, sparse digraph. * Research supported by FQRNT (Le Fonds québécois de la recherche sur la nature et les technologies) doctoral scholarship. †

The Barát-Thomassen conjecture asserts that for every tree T on m edges, there exists a constant k T such that every k T -edge-connected graph with size divisible by m can be edge-decomposed into copies of T . So far this conjecture has only been verified when T is a path or when T has diameter at most 4. Here we prove the full statement of the conjecture. condition is also sufficient in certain cases. By a result of Wilson [17] this holds when G is a sufficiently large complete graph, and there exist more general results showing that this is also true for graphs of large minimum degree. More precisely, for every tree T there exists a constant ε T > 0 such that every graph G of minimum degree (1 − ε T )|V (G)| admits a T -decomposition, provided its size is divisible by the size of T (see for example [2]).A different line of research was started by Barát and Thomassen [3], when they observed in 2006 that T -decompositions are intimately related to nowhere-zero flows. Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow, but until recently it was not even known that any constant edge-connectivity suffices for this. Barát and Thomassen showed that if every 8-edge-connected graph of size divisible by 3 admits a K 1,3 -decomposition, then every 8-edge-connected graph admits a nowhere-zero 3-flow. Vice versa, they also showed that Tutte's 3-flow conjecture would imply that every 10-edgeconnected graph with size divisible by 3 admits a K 1,3 -decomposition. Motivated by this intrinsic connection, they conjectured the following.Conjecture 1 (Barát-Thomassen Conjecture, [3]). For any tree T on m edges, there exists an integer k T such that every k T -edge-connected graph with size divisible by m has a Tdecomposition.When the conjecture was made, it was only known to hold in the trivial cases where T has less than 3 edges. Since then, Conjecture 1 has attracted growing attention, and it has now been verified for different families of trees such as stars [14], some bistars [1,16], and paths of a certain length [6,12,13,15]. Very recently, breakthrough results were obtained by Merker [10], who proved the conjecture for all trees of diameter at most 4, hence covering some of the results above, and by Botler, Mota, Oshiro, and Wakabayashi [5], who proved the conjecture for all paths. The latter result was improved by Bensmail, Harutyunyan, Le, and Thomassé [4], who showed that, for path-decompositions, large minimum degree is a sufficient condition provided the graph is 24-edge-connected.The purpose of this paper is to verify Conjecture 1 for every tree T , hence settling the conjecture in the affirmative.Theorem 2. The Barát-Thomassen conjecture is true.This paper builds upon previous work on the Barát-Thomassen conjecture. It was shown by Thomassen in [16], and independently by Barát and Gerbner in [1], that it is sufficient to verify Conjecture 1 for bipartite graphs G.Theorem 3. [1,16] Let T be a tree on m edges. The following two statements are equivalent:(1) There exists a natural number k T...

Abstract. Let C and D be digraphs. A mapping f :, and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively Ccolourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number r ≥ 1, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.

In 2006, Barát and Thomassen conjectured that there is a function f such that, for every fixed tree T with t edges, every f (t)-edgeconnected graph with its number of edges divisible by t has a partition of its edges into copies of T . This conjecture was recently verified by the current authors and Merker [1].We here further focus on the path case of the Barát-Thomassen conjecture. Before the aforementioned general proof was announced, several successive steps towards the path case of the conjecture were made, notably by Thomassen [11,12,13], until this particular case was totally solved by Botler, Mota, Oshiro and Wakabayashi [2]. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function f such that every 24-edge-connected graph with minimum degree f (t) has an edge-partition into paths of length t whenever t divides the number of edges. We also show that 24 can be dropped to 4 when the graph is eulerian.

Neumann-Lara (1985) andŠkrekovski conjectured that every planar digraph with digirth at least three is 2-colorable, meaning that the vertices can be 2-colored without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least five is 2-colorable. The result also holds in the setting of list colorings.