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 G1, G2, . . . , 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 ax = by implies a = b and x = y whenever x, y ∈ S and 1 ≤ a, b ≤ k. Suppose that χ(Gi, F) ≤ k and S is a k-multiplicative Sidon set of cardinality d. We prove that χ(H, F) ≤ 1+2k•max 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(dk log k). We illustrate the method with numerous examples, some of which generalise or improve upon existing results in the literature.

Coloring the square of the Cartesian product of two cycles

“…Our main result will improve the bounds provided by Theorems 1 and 2 and by the general result of Pór and Wood [13]:…”

“…As in the proof of Theorem 6, we use combinations of K and K 3 (the corresponding pattern, not the complete graph) to obtain an m × 13 pattern X for m ∈ S(7, 3). We then use combinations of X and X ′ 4 to obtain an m × n pattern for n ∈ S (13,4). In this way, we can obtain proper 7-colorings of T 2 13,13 , T 2 17,13 , and T 2 17,17 .…”

“…A proper coloring of the square G 2 of a graph G is often called a distance-2 coloring of G. In [13], Pór and Wood studied the notion of F -free coloring. Let F be a family of connected bipartite graphs, each with at least three vertices.…”

The 2-Ranking Numbers of Graphs

“…, and improved the upper bound to 2d + 1 in the case that 2d + 1 divides the length of each factor cycle. Pór and Wood [8] proved that G admits a proper (6d + O(log d))coloring in which each pair of color classes induces a matching and isolated vertices; their result directly implies that χ s (G) ≤ 6d + O(log d). Corollary 3 extends the divisibility conditions under which it is known that χ s (G) ≤ 2d + 1.…”

Game coloring the Cartesian product of graphs

“…It is proved in [16] that the Cartesian product of d trees has acyclic chromatic number at most d + 1. It follows from a result in [25] that the Cartesian product of d planar graphs has acyclic chromatic number at most 50(d − 1) − 9. The product of two planar graphs has acyclic chromatic number at most 25.…”