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 …

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We prove a conjecture of Bonamy, Bousquet, Pilipczuk, Rz\k {a}\. zewski, Thomass\'e, and
Walczak, that for every graph $ H $, there is a polynomial $ p $ such that for every positive …

In this paper we prove that for every $ s\geq 2$ and every graph $ H $ the following holds.
Let $ G $ be a graph with average degree $\Omega_H (s^{C| H|^ 2}) $, for some absolute …

A hereditary class of graphs G is χ-bounded if there exists a function f such that every graph
G∈ G satisfies χ (G)⩽ f (ω (G)), where χ (G) and ω (G) are the chromatic number and the …

Treewidth is an important graph invariant, relevant for both structural and algorithmic
reasons. A necessary condition for a graph class to have bounded treewidth is the absence …

Let p and q be positive integers with p/q≥ 2. The “reconfiguration problem” for circular
colourings asks, given two (p, q)-colourings f and g of a graph G, is it possible to transform f …

A hole is an induced cycle of length at least 4. Let $\l\ge 2$ be a positive integer, let ${\cal G}
_l $ denote the family of graphs which have girth $2\l+ 1$ and have no holes of odd length at …

A large body of research in graph theory concerns the induced subgraphs of graphs with
large chromatic number, and especially which induced cycles must occur. In this paper, we …