Vol. 11, No. 2, 2018

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Forbidden subgraphs of coloring graphs

Francisco Alvarado, Ashley Butts, Lauren Farquhar and Heather M. Russell

Vol. 11 (2018), No. 2, 311–324

Given a graph G, its k-coloring graph has vertex set given by the proper k-colorings of the vertices of G with two k-colorings adjacent if and only if they differ at exactly one vertex. Beier et al. (Discrete Math. 339:8 (2016), 2100–2112) give various characterizations of coloring graphs, including finding graphs which never arise as induced subgraphs of coloring graphs. These are called forbidden subgraphs, and if no proper subgraph of a forbidden subgraph is forbidden, it is called minimal forbidden. In this paper, we construct a finite collection of minimal forbidden subgraphs that come from modifying theta graphs. We also construct an infinite family of minimal forbidden subgraphs similar to the infinite family found by Beier et al.

proper graph coloring, coloring graph, forbidden subgraph
Mathematical Subject Classification 2010
Primary: 05C15
Received: 10 November 2016
Revised: 14 May 2017
Accepted: 19 June 2017
Published: 17 September 2017

Communicated by Jerrold Griggs
Francisco Alvarado
California State University
Los Angeles, CA
United States
Ashley Butts
University of the Pacific
Stockton, CA
United States
Lauren Farquhar
Department of Mathematics
University of Colorado
Boulder, CO
United States
Heather M. Russell
Department of Mathematics
University of Richmond
Richmond, VA
United States