If a graph has bounded clique number and sufficiently large chromatic number, what can we
say about its induced subgraphs? András Gyárfás made a number of challenging …

A graph G with clique number ω (G) ω (G) and chromatic number χ (G) χ (G) is perfect if χ
(H)= ω (H) χ (H)= ω (H) for every induced subgraph H of G. A family GG of graphs is called χ …

Treewidth is a parameter that emerged from the study of minor closed classes of graphs (ie
classes closed under vertex and edge deletion, and edge contraction). It in some sense …

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a
cycle of length at least 5 with exactly one chord and that chord creates a triangle with the …

The class of even-hole-free graphs is structurally quite similar to the class of perfect graphs,
which was the key initial motivation for their study. The techniques developed in the study of …

We prove that the tree independence number of every even-hole-free graph is at most
polylogarithmic in its number of vertices. More explicitly, we prove that there exists a …

We elucidate the structure of (P_6,C_4)-free graphs by showing that every such graph either
has a clique cutset, or a universal vertex, or belongs to several special classes of graphs …

The three-in-a-tree problem is to determine if a simple undirected graph contains an induced
subgraph which is a tree connecting three given vertices. Based on a beautiful …

In this paper we show that every (P 6, diamond)‐free graph G satisfies χ (G)≤ ω (G)+ 3,
where χ (G) and ω (G) are the chromatic number and clique number of G, respectively. Our …

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 …