Vol. 12, No. 7, 2019

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On the Hadwiger number of Kneser graphs and their random subgraphs

Arran Hamm and Kristen Melton

Vol. 12 (2019), No. 7, 1153–1161
Abstract

For n,k , let KG(n,k) be the usual Kneser graph (whose vertices are k-sets of {1,2,,n} with A B if and only if A B = ). The Hadwiger number of a graph G, denoted by h(G), is max{t : Kt G}, where H G if H is a minor of G. Previously, lower bounds have been given on the Hadwiger number of a graph in terms of its average degree. In this paper we give lower bounds on h(KG(n,k)) and h(KG(n,k)p), where KG(n,k)p is the binomial random subgraph of KG(n,k) with edge probability p. Each of these bounds is larger than previous bounds under certain conditions on k and p.

Keywords
Kneser graphs, Hadwiger number
Mathematical Subject Classification 2010
Primary: 05C83, 05C80, 05D40
Milestones
Received: 31 October 2018
Revised: 19 February 2019
Accepted: 11 May 2019
Published: 12 October 2019

Communicated by Anant Godbole
Authors
Arran Hamm
Department of Mathematics
Winthrop University
Rock Hill, SC
United States
Kristen Melton
Department of Mathematics
Miami University
Oxford, OH
United States