# Connectivity and Hamiltonicity of Canonical Colouring Graphs of Bipartite and Complete Multipartite Graphs

^{1}

Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI 96822, USA

^{2}

Smith College, Northampton, MA 01063, USA

^{3}

Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 2Y2, Canada

^{*}

Author to whom correspondence should be addressed.

Received: 12 February 2018 / Revised: 23 March 2018 / Accepted: 24 March 2018 / Published: 29 March 2018

(This article belongs to the Special Issue Reconfiguration Problems)

# Abstract

A k-colouring of a graph G with colours $1,2,\dots ,k$ is canonical with respect to an ordering $\pi ={v}_{1},{v}_{2},\dots ,{v}_{n}$ of the vertices of G if adjacent vertices are assigned different colours and, for $1\le c\le k$ , whenever colour c is assigned to a vertex ${v}_{i}$ , each colour less than c has been assigned to a vertex that precedes ${v}_{i}$ in $\pi $ . The canonical k-colouring graph of G with respect to π is the graph ${\mathrm{Can}}_{k}^{\pi}\left(G\right)$ with vertex set equal to the set of canonical k-colourings of G with respect to $\pi $ , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering $\pi $ such that ${\mathrm{Can}}_{k}^{\pi}\left(G\right)$ is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering $\pi $ such that ${\mathrm{Can}}_{k}^{\pi}\left({K}_{m,n}\right)$ has a Hamilton path for all $k\ge 3$ . The paper concludes with a detailed consideration of ${\mathrm{Can}}_{k}^{\pi}\left({K}_{2,2,\dots ,2}\right)$ . For each $k\ge \chi $ and all vertex orderings $\pi $ , it is proved that ${\mathrm{Can}}_{k}^{\pi}\left({K}_{2,2,\dots ,2}\right)$ is either disconnected or isomorphic to a particular tree. View Full-Text*Keywords:*reconfiguration problems; graph colouring; Hamilton cycles; Gray codes

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**MDPI and ACS Style**

Haas, R.; MacGillivray, G. Connectivity and Hamiltonicity of Canonical Colouring Graphs of Bipartite and Complete Multipartite Graphs. *Algorithms* **2018**, *11*, 40.

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