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On minimal Ramsey graphs and Ramsey equivalence in multiple colours

Published online by Cambridge University Press:  09 March 2020

Dennis Clemens
Affiliation:
Institut für Mathematik, Technische Universität Hamburg, 21073Hamburg, Germany
Anita Liebenau*
Affiliation:
School of Mathematics and Statistics, UNSW Sydney, SydneyNSW2052, Australia
*
*Corresponding author. Email: a.liebenau@unsw.edu.au

Abstract

For an integer q ⩾ 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H yet no proper subgraph of G has this property, then G is called q-Ramsey-minimal for H. Generalizing a statement by Burr, Nešetřil and Rödl from 1977, we prove that, for q ⩾ 3, if G is a graph that is not q-Ramsey for some graph H, then G is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H as long as H is 3-connected or isomorphic to the triangle. For such H, the following are some consequences.

  • For 2 ⩽ r < q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H.

  • For every q ⩾ 3, there are q-Ramsey-minimal graphs for H of arbitrarily large maximum degree, genus and chromatic number.

  • The collection $\{\mathcal M_q(H) \colon H \text{ is 3-connected or } K_3\}$ forms an antichain with respect to the subset relation, where $\mathcal M_q(H)$ denotes the set of all graphs that are q-Ramsey-minimal for H.

We also address the question of which pairs of graphs satisfy $\mathcal M_q(H_1)=\mathcal M_q(H_2)$ , in which case H1 and H2 are called q-equivalent. We show that two graphs H1 and H2 are q-equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q ⩾ 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.

MSC classification

Type
Paper
Copyright
© Cambridge University Press 2020

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Footnotes

Supported by an ARC Decra Fellowship. Previously at Monash University where this research was partly carried out.

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