## 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

#### Abstract

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

**Source**

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

**Dates**

Received: 10 November 2016

Revised: 14 May 2017

Accepted: 19 June 2017

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513775065

**Digital Object Identifier**

doi:10.2140/involve.2018.11.311

**Mathematical Reviews number (MathSciNet)**

MR3733960

**Zentralblatt MATH identifier**

06817023

**Subjects**

Primary: 05C15: Coloring of graphs and hypergraphs

**Keywords**

proper graph coloring coloring graph forbidden subgraph

#### Citation

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. https://projecteuclid.org/euclid.involve/1513775065