Graph classes with linear Ramsey numbers



Alecu, Bogdan, Atminas, Aistis, Lozin, Vadim and Zamaraev, Viktor ORCID: 0000-0001-5755-4141
(2021) Graph classes with linear Ramsey numbers. Discrete Mathematics, 344 (4).

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Abstract

The Ramsey number $R_X(p,q)$ for a class of graphs $X$ is the minimum $n$ such that every graph in $X$ with at least $n$ vertices has either a clique of size $p$ or an independent set of size $q$. We say that Ramsey numbers are linear in $X$ if there is a constant $k$ such that $R_{X}(p,q) \leq k(p+q)$ for all $p,q$. In the present paper we conjecture that if $X$ is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in $X$ if and only if $X$ excludes a forest, a disjoint union of cliques and their complements. We prove the "only if" part of this conjecture and verify the "if" part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case.

Item Type: Article
Additional Information: 24 pages
Uncontrolled Keywords: math.CO, math.CO
Depositing User: Symplectic Admin
Date Deposited: 08 Mar 2021 10:52
Last Modified: 11 Aug 2022 11:27
DOI: 10.1016/j.disc.2021.112307
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3063604