van den Heuvel, Jan, Kierstead, H. A and Quiroz, Daniel
(2019)
Chromatic numbers of exact distance graphs.
Journal of Combinatorial Theory, Series B, 134.
pp. 143163.
ISSN 00958956
Abstract
For any graph G = (V;E) and positive integer p, the exact distancep graph G[\p] 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, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[\p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil 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[\p] is bounded by the weak (2p
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