Colourings of the Cartesian Product of Graphs and Multiplicative Sidon Sets

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 Gi and the following number-theoretic concept. A set S of natural numbers is k-multiplicative Sidon if a x = b y implies a = b and x = y whenever x , y ∈ S and 1 ⩽ a , b ⩽ k . 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 + 2 k ⋅ m a x S . We then prove that the maximum density of a k-multiplicative Sidon set is Θ ( 1 / log k ) . It follows that χ ( H , F ) ⩽ O ( d k log k ) .