Colourings of the cartesian product of graphs and multiplicative Sidon sets

Abstract

Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is F-free. The F-free chromatic number χ(G,F) of a graph G is the minimum number of colours in an F-free colouring of G. For appropriate choices of F, several well-known types of colourings fit into this framework, including acyclic colourings, star colourings, and distance-2 colourings. This paper studies F-free colourings of the cartesian product of graphs.

Let H be the cartesian product of the graphs G 1,G 2,…,G d . Our main result establishes an upper bound on the F-free chromatic number of H in terms of the maximum F-free chromatic number of the G i and the following number-theoretic concept. A set S of natural numbers is k-multiplicative Sidon if ax=by implies a=b and x=y whenever x,yS and 1≤a, bk. Suppose that χ(G i ,F)≤k and S is a k-multiplicative Sidon set of cardinality d. We prove that χ(H,F)≤1+2k·maxS. We then prove that the maximum density of a k-multiplicative Sidon set is gJ(1/log k). It follows that χ(H,F)≤\( \mathcal{O} \)(dk logk). We illustrate the method with numerous examples, some of which generalise or improve upon existing results in the literature.

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Correspondence to Attila Pór.

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Research initiated at the Department of Applied Mathematics and the Institute for Theoretical Computer Science, Charles University, Prague, Czech Republic. Supported by project LN00A056 of the Ministry of Education of the Czech Republic, and by the European Union Research Training Network COMBSTRU (Combinatorial Structure of Intractable Problems).

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Pór, A., Wood, D.R. Colourings of the cartesian product of graphs and multiplicative Sidon sets. Combinatorica 29, 449–466 (2009). https://doi.org/10.1007/s00493-009-2257-0

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Mathematics Subject Classification (2000)

  • 05C15
  • 11N99