Hadwiger's Conjecture and inflations of the Petersen graph

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Abstract

An inflation of a graph G is obtained by replacing vertices in G by disjoint cliques and adding all possible edges between any pair of cliques corresponding to adjacent vertices in G. We prove that the chromatic number of an arbitrary inflation F of the Petersen graph is equal to the chromatic number of some inflated 5-cycle contained in F. As a corollary, we find that Hadwiger's Conjecture holds for any inflation of the Petersen graph. This solves a problem posed by Bjarne Toft. © 2012 Elsevier B.V. All rights reserved.

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Pedersen, A. S. (2012). Hadwiger’s Conjecture and inflations of the Petersen graph. Discrete Mathematics, 312(24), 3537–3543. https://doi.org/10.1016/j.disc.2012.08.007

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