The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, K 4 )-free graphs was constructed, that still has unbounded tree-width [Sintiari and Trotignon, 2019]. The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary.We prove that for every graph G, if G excludes a fixed graph H as a minor, then G either has small tree-width, or G contains a large wall or the line graph of a large wall as induced subgraph. This can be seen as a strengthening of Robertson and Seymour's excluded grid theorem for the case of minor-free graphs. Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width. In fact, our theorem applies to a more general class: (theta, prism)-free graphs. This implies the known result that planar even hole-free graph have bounded tree-width [da Silva and Linhares Sales, Discrete Applied Mathematics 2010].We conjecture that even-hole-free graphs of bounded degree have bounded tree-width. If true, this would mean that even-hole-freeness is testable in the bounded-degree graph model of property testing. We prove the conjecture for subcubic graphs and we give a bound on the

The dichromatic number of a digraph D is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been a recent center of study. In this work we look at possible extensions of Gyárfás-Sumner conjecture. More precisely, we propose as a conjecture a simple characterization of finite sets F of digraphs such that every oriented graph with sufficiently large dichromatic number must contain a member of F as an induce subdigraph.Among notable results, we prove that oriented triangle-free graphs without a directed path of length 3 are 2-colorable. If condition of "triangle-free" is replaced with "K 4 -free", then we have an upper bound of 414. We also show that an orientation of complete multipartite graph with no directed triangle is 2-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

The urethral syndrome is a frequently encountered psychosomatic disorder; it constitutes the main complaint of some 20% of all women who consult a urologist. After a brief survey of the history of this syndrome, the authors examine 55 case reports of women with the urethral syndrome. The authors particularly emphasize the significance of the manner in which patients describe their perceptions of the pain, which is a valuable diagnostic sign. A follow-up study of patients for 3 to 21 years after the first interview demonstrated the releative efficiency of the various forms of treatment. A discussion of the therapeutic possibilities and their limitations follows.

The dichromatic number χ(D) of a digraph D is the least integer k such that D can be partitionedAn oriented graph is a digraph with no directed cycle of length 2. For integers k and n, we denote by o k (n) the minimum number of edges of a k-critical oriented graph on n vertices (with the convention o k (n) = +∞ if there is no k-dicritical oriented graph of order n). The main result of this paper is a proof that o 3 (n) ≥ 7n+2 3 together with a construction witnessing that o 3 (n) ≤ 5n 2 for all n ≥ 12. We also give a construction showing that for all sufficiently large n and all k ≥ 3, o k (n) < (2k − 3)n, disproving a conjecture of Hoshino and Kawarabayashi. Finally, we prove that, for all k ≥ 2, o k (n) ≥ k − 3 4 − 1 4k−6 n + 3 4(2k−3) , improving the previous best known lower bound of Bang-Jensen, Bellitto, Schweser and Stiebitz.