Involve: A Journal of Mathematics

  • Involve
  • Volume 11, Number 2 (2018), 311-324.

Forbidden subgraphs of coloring graphs

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

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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.

Article information

Involve, Volume 11, Number 2 (2018), 311-324.

Received: 10 November 2016
Revised: 14 May 2017
Accepted: 19 June 2017
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C15: Coloring of graphs and hypergraphs

proper graph coloring coloring graph forbidden subgraph


Alvarado, Francisco; Butts, Ashley; Farquhar, Lauren; Russell, Heather M. Forbidden subgraphs of coloring graphs. Involve 11 (2018), no. 2, 311--324. doi:10.2140/involve.2018.11.311.

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  • J. Beier, J. Fierson, R. Haas, H. M. Russell, and K. Shavo, “Classifying coloring graphs”, Discrete Math. 339:8 (2016), 2100–2112.
  • M. Cohen and M. Teicher, “Kauffman's clock lattice as a graph of perfect matchings: a formula for its height”, Electron. J. Combin. 21:4 (2014), art. id. 4.31.
  • R. Diestel, Graph theory, Graduate Texts in Mathematics 173, Springer, New York, 1997.
  • M. Dyer, A. D. Flaxman, A. M. Frieze, and E. Vigoda, “Randomly coloring sparse random graphs with fewer colors than the maximum degree”, Random Structures Algorithms 29:4 (2006), 450–465.
  • R. Haas, “The canonical coloring graph of trees and cycles”, Ars Math. Contemp. 5:1 (2012), 149–157.
  • M. Jerrum, “A very simple algorithm for estimating the number of $k$-colorings of a low-degree graph”, Random Structures Algorithms 7:2 (1995), 157–165.
  • B. Mohar, “Kempe equivalence of colorings”, pp. 287–297 in Graph theory in Paris, edited by A. Bondy et al., Birkhäuser, Basel, 2007.
  • M. Molloy, “The Glauber dynamics on colorings of a graph with high girth and maximum degree”, SIAM J. Comput. 33:3 (2004), 721–737.
  • E. Vigoda, “Improved bounds for sampling colorings”, J. Math. Phys. 41:3 (2000), 1555–1569.
  • F. J. Zhang, X. F. Guo, and R. S. Chen, “$Z$-transformation graphs of perfect matchings of hexagonal systems”, Discrete Math. 72:1-3 (1988), 405–415.