A Reconfigurations Analogue of Brooks’ Theorem

Feghali, C.; Johnson, M.; Paulusma, D.

Abstract

Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless G is a complete graph or a cycle with an odd number of vertices, or k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike. We use this result to study the reconfiguration graph R k (G) of the k-colourings of G. The vertex set of R k (G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that if k ≤ Δ(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter, if k ≥ Δ(G) + 2, then R k (G) is connected and has diameter O(n 2). We complete this structural classification by settling the missing case: if k = Δ(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2). We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R k (G) by settling the case k = Δ(G) + 1. The problem is O(n 2) time solvable for k = 3, PSPACE-complete for 4 ≤ k ≤ Δ(G), O(n) time solvable for k = Δ(G) + 1, O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes).

Citation

Feghali, C., Johnson, M., & Paulusma, D. (2014, December). A Reconfigurations Analogue of Brooks’ Theorem. Presented at 39th International Symposium, MFCS 2014, Budapest, Hungary

• Presentation Conference Type: Conference Paper (published)
• Conference Name: 39th International Symposium, MFCS 2014
• Publication Date: Jan 1, 2014
• Deposit Date: Dec 20, 2014
• Publicly Available Date: Jan 16, 2015
• Print ISSN: 0302-9743
• Pages: 287-298
• Series Title: Lecture notes in computer science
• Series Number: 8635
• Series ISSN: 0302-9743,1611-3349
• Book Title: 39th International Symposium, MFCS 2014, Budapest, Hungary, 26-29 August 2014 ; proceedings, Part II.
• ISBN: 9783662444641
• DOI: https://doi.org/10.1007/978-3-662-44465-8_25
• Public URL: https://durham-repository.worktribe.com/output/1153498