A hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An even-hole-free graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. In this paper we prove that every even-hole-free graph has a bisimplicial vertex, which was originally conjectured by Reed.

Abstract:A graph G has maximal local edge-connectivity k if the maximum number of edge-disjoint paths between every pair of distinct vertices x and y is at most k. We prove Brooks-type theorems for k-connected graphs with maximal local edge-connectivity k, and for any graph with maximal local edge-connectivity 3. We also consider several related graph classes defined by constraints on connectivity. In particular, we show that there is a polynomial-time algorithm that, given a 3-connected graph G with maximal local connectivity 3, outputs an optimal coloring for G. On the other hand, we prove, for k ≥ 3, that k-COLORABILITY is NP-complete when restricted to minimally k-connected graphs, and 3-COLORABILITY is NPcomplete when restricted to (k − 1)-connected graphs with maximal local connectivity k. Finally, we consider a parameterization of k-COLORABILITY based on the number of vertices of degree at least k + 1, and prove that, even when k is part of the input, the corresponding parameterized problem is FPT. C 2016 Wiley Periodicals, Inc. J. Graph Theory 85: 2017

In 1985, Mader conjectured the existence of a function f such that every digraph with minimum out-degree at least f (k) contains a subdivision of the transitive tournament of order k. This conjecture is still completely open, as the existence of f (5) remains unknown. In this paper, we show that if D is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from x to y and a directed path from y to x, then every digraph with minimum out-degree large enough contains a subdivision of D. Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph D is the smallest integer k such that D can be partitioned into k acyclic subdigraphs. We show that any digraph with dichromatic number greater than 4 m (n − 1) contains every digraph with n vertices and m arcs as a subdivision.Conjecture 2 (Mader [20]). There exists a least integer mader δ + (T T k ) such that every digraph D with δ + (D) ≥ mader δ + (T T k ) contains a subdivision of T T k .Mader proved that mader δ + (T T 4 ) = 3, but even the existence of mader δ + (T T 5 ) is still open. This conjecture implies directly that transitive tournaments (and thus all acyclic digraphs) are δ 0maderian. Conjecture 3. There exists a least integer maderIn fact, Conjecture 3 is equivalent to Conjecture 2 because if transitive tournaments are δ 0 -maderian, then mader δ + (T T k ) ≤ mader δ 0 (T T 2k ) for all k. Indeed, let D be a digraph with minimum out-degree mader δ 0 (T T 2k ), and let D ′ be the digraph obtained from disjoint copies of D and its converse (the digraph obtained by reversing all arcs) D by adding all arcs from D to D. Clearly, δ 0 (D ′ ) ≥ mader δ 0 (T T 2k ). Therefore D ′ contains a subdivision of T T 2k . Hence, either D or D (and so D) contains a subdivision of T T k .Both conjectures are equivalent, but the above reasoning does not prove that a δ 0 -maderian digraph is also δ + -maderian. The case of oriented trees (i.e. orientations of undirected trees) is typical. Using a simple greedy procedure, one can easily find every oriented tree of order k in every digraph with minimum in-and out-degree k (so mader δ 0 (T ) = |T | − 1 for any oriented tree T ). On the other hand, it is still open whether oriented trees are δ + -maderian and a natural important step towards Conjecture 2 would be to prove the following weaker one.Conjecture 4. Every oriented tree is δ + -maderian.We give evidences to this conjecture. First, in Subsection 2.1, we prove that every oriented path (i.e. orientation of an undirected path) P is δ + -maderian and that mader δ + (P ) = |V (P )| − 1. Next, in Subsection 2.2, we consider arborescences. An out-arborescence (resp. in-arborescence) is an oriented tree in which all arcs are directed away from (resp. towards) a vertex called the root. Trivially, the simple greedy procedure shows that mader δ + (T ) = |T | − 1 for every out-arborescence. In contrast, the fact that in-arborecences are δ + -maderian is no...

Improper choosability of planar graphs has been widely studied. In particular,Škrekovski investigated the smallest integer g k such that every planar graph of girth at least g k is k-improper 2-choosable. He proved [9] that 6≤ g 1 ≤9; 5≤ g 2 ≤ 7; 5 ≤ g 3 ≤ 6; and ∀k ≥ 4, g k = 5. In this article, we study the greatest real M(k, l) such that every graph of maximum average degree less than. As a corollary, we deduce that g 1 ≤ 8 and g 2 ≤ 6, and we obtain new results for graphs of higher genus. We also provide an upper bound for M (k, l). This implies that for any fixed l, M(k, l) −→ k→∞ 2l.