Hadwiger's Conjecture for inflations of 3-chromatic graphs

Hadwiger’s Conjecture states that every k-chromatic graph has a complete minor of order k. A graph G is an inflation of a graph G if G is obtained from G by replacing each vertex v of G by a clique Cv and joining two vertices of distinct cliques by an edge if and only if the corresponding vertices of G are adjacent. We present an algorithm for computing an upper bound on the chromatic number χ(G) of any inflation G of any 3-chromatic graph G. As a consequence, we deduce that Hadwiger’s Conjecture holds for any inflation of any 3-colorable graph. Keywords: Hadwiger’s Conjecture, graph coloring, inflation, 3-chromatic graph, complete minor 1 Introduction A proper k-coloring of a graph G is a function f ∶ V (G) → {1, . . . , k} such that f(v) ≠ f(u) whenever u and v are adjacent. The chromatic number χ(G) of G is the smallest k such that there is a proper k-coloring of G. A graph G is k-chromatic if χ(G) = k. Hadwiger’s Conjecture is one of the fundamental open questions in graph coloring. It dates back to 1943, when Hadwiger [7] suggested that every k-chromatic graph G contains a complete minor of order k, i.e. a complete graph of order k can be obtained from G by deleting and/or contracting edges. The conjecture is a far-reaching generalization of the well-known Four Color Problem, which asks if every planar graph has chromatic number at most 4, and it remains open for all k greater than 6. (See [15] for a survey on Hadwiger’s Conjecture.) The case k ≤ 4 was proved by Hadwiger in his original paper [7]. Wagner [16] proved that the case k = 5 is equivalent to the Four Color Problem. The latter problem was solved in the affirmative by Appel and Haken [1, 2] in 1977, and in 1993 Robertson et al. [12] proved Hadwiger’s Conjecture for k = 6. ∗Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden. E-mail address: [email protected] Part of the work done while the author was a postdoc at University of Southern Denmark and at Mittag-Leffler Institute. Research supported by SVeFUM and Mittag-Leffler Institute. „Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense, Denmark. E-mail address: [email protected] …Research Clinic on Gambling Disorders, Center for Functionally Integrative Neuroscience, Aarhus University Hospital, Trøjborgvej 72, Bygning 30, 8200 Aarhus N, Denmark.