Given a graph H, the Ramsey number r (H) is the smallest natural number N such that any
two-colouring of the edges of KN contains a monochromatic copy of H. The existence of …

The regularity lemma of Szemerédi asserts that every graph can be partitioned in a certain
regular way. This result has numerous applications, but its known proof is not algorithmic …

We show that the game chromatic number of a planar graph is at most 33. More generally,
there exists a function f: ℕ→ ℕ so that for each n∈ ℕ, if a graph does not contain a …

We describe a simple and yet surprisingly powerful probabilistic technique which shows
how to find in a dense graph a large subset of vertices in which all (or almost all) small …

For a graph H and an integer r≥ 2, the induced r-size-Ramsey number of H is defined to be
the smallest integer m for which there exists a graph G with m edges with the following …

Given a graph G and a positive integer p, χp (G) is the minimum number of colours needed
to colour the vertices of G so that for any i≤ p, any subgraph H of G of tree-depth i gets at …

The k-domination number of a graph is the minimum size of a set D such that every vertex of
G is at distance at most k from D. We give a linear-time constant-factor algorithm for …

The odd chromatic number and the conflict-free chromatic number are new graph
parameters introduced by Petru\v {s} evski and\v {S} krekovski [2021] and Fabrici, Lu\v {z} ar …

In this paper, we present several density-type theorems which show how to find a copy of a
sparse bipartite graph in a graph of positive density. Our results imply several new bounds …

The planar graph product structure theorem of Dujmović et al.[J. ACM, 67 (2020), 22] states
that every planar graph is a subgraph of the strong product of a graph with bounded …