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http://dx.doi.org/10.5666/KMJ.2016.56.3.647

Reconfiguring k-colourings of Complete Bipartite Graphs  

Celaya, Marcel (School of Mathematics, Georgia Tech)
Choo, Kelly (Mathematics and Statistics, University of Victoria)
MacGillivray, Gary (Mathematics and Statistics, University of Victoria)
Seyffarth, Karen (Mathematics and Statistics, University of Calgary)
Publication Information
Kyungpook Mathematical Journal / v.56, no.3, 2016 , pp. 647-655 More about this Journal
Abstract
Let H be a graph, and $k{\geq}{\chi}(H)$ an integer. We say that H has a cyclic Gray code of k-colourings if and only if it is possible to list all its k-colourings in such a way that consecutive colourings, including the last and the first, agree on all vertices of H except one. The Gray code number of H is the least integer $k_0(H)$ such that H has a cyclic Gray code of its k-colourings for all $k{\geq}k_0(H)$. For complete bipartite graphs, we prove that $k_0(K_{\ell},r)=3$ when both ${\ell}$ and r are odd, and $k_0(K_{\ell},r)=4$ otherwise.
Keywords
reconfiguration problems; graph colouring; Hamilton cycles; Gray codes;
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