Chromatic number for a generalization of cartesian product graphs

Abstract

Let G be a class of graphs. A d-fold grid over G is a graph obtained from a d-dimensional rectangular grid of vertices by placing a graph from G on each of the lines parallel to one of the axes. Thus each vertex belongs to d of these subgraphs. The class of d-fold grids over G is denoted by Gd. Let f(G; d) = maxG∈Gd χ(G). If each graph in G is k-colorable, then f(G; d) ≤ κd. We show that this bound is best possible by proving that f(G; d) = κd when G is the class of all k-colorable graphs. We also show that f(G; d) ≥⌊ √d/6 log d ⌋when G is the class of graphs with at most one edge, and f(G; d) ≥⌊ √d/6 log d ⌋ when G is the class of graphs with maximum degree 1.