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...

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vskovi\'c. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-hole-free graphs, square-theta-free Berge graphs, universally signable graphs and wheel-free graphs. Some other classes are new. It turns out that all the classes that we study in this paper can be defined by excluding some of the so-called Truemper configurations. For several classes of graphs, we obtain directly bounds on the chromatic number, or fast algorithms for the maximum clique problem or the coloring problem

A classic theorem of Euclidean geometry asserts that any noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space.We prove that in any metric space with n points, either there is a line containing all the points or there are at least Ω( √ n) lines. This is the first polynomial lower bound on the number of lines in general finite metric spaces. In the more general setting of pseudometric betweenness, we prove a corresponding bound of Ω(n 2/5 ) lines. When the metric space is induced by a connected graph, we prove that either there is a line containing all the points or there are Ω(n 4/7 ) lines, improving the previous Ω(n 2/7 ) bound. We also prove that the number of lines in an n-point metric space is at least n/5w, where w is the number of different distances in the space, and we give an Ω(n 4/3 ) lower bound on the number of lines in metric spaces induced by graphs with constant diameter, as well as spaces where all the positive distances are from {1, 2, 3}.