Chromatic Numbers of Exact Distance Graphs

For any graph G = (V, E) and positive integer p, the exact distance-p graph G([tP]) is the graph with vertex set V, which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nesetril and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G([tp]) is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nesetril and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G([tP]) is bounded by the weak (2p - 1)-colouring number of G. For even p, we prove that chi(G([tP])) is at most the weak (2p)-colouring number times the maximum degree. For odd p, the existing lower bound on the number of colours needed to colour G([tP]) when G is planar is improved. Similar lower bounds are given for K-t-minor free graphs. (C) 2018 Elsevier Inc. All rights reserved.