research-article

A polynomial version of Cereceda's conjecture

Authors:

Nicolas Bousquet,

Marc HeinrichAuthors Info & Claims

Volume 155, Issue C

Pages 1 - 16

https://doi.org/10.1016/j.jctb.2022.01.006

Published: 01 July 2022 Publication History

Abstract

Let k and d be positive integers such that k ≥ d + 2. Consider two k-colourings of a d-degenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper colouring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length.

The k-reconfiguration graph of G is the graph whose vertices are the proper k-colourings of G, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the ( d + 2 )-reconfiguration graph of any d-degenerate graph on n vertices is O ( n 2 ). So far, there is no proof of a polynomial upper bound on the diameter, even for d = 2.

In this paper, we prove that the diameter of the k-reconfiguration graph of a d-degenerate graph is O ( n d + 1 ) for k ≥ d + 2. Moreover, we prove that if k ≥ 3 2 ( d + 1 ) then the diameter of the k-reconfiguration graph is quadratic, improving the previous bound of k ≥ 2 d + 1. We also show that the 5-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs.

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Cited By

Cambie SCames van Batenburg WCranston D(2024)Optimally reconfiguring list and correspondence colouringsEuropean Journal of Combinatorics10.1016/j.ejc.2023.103798115:COnline publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1016/j.ejc.2023.103798

Index Terms

A polynomial version of Cereceda's conjecture
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
      2. Graph enumeration
2. Theory of computation
  1. Randomness, geometry and discrete structures

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Journal of Combinatorial Theory Series B

Journal of Combinatorial Theory Series B Volume 155, Issue C

Jul 2022

389 pages

ISSN:0095-8956

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Elsevier Inc.

Publisher

Academic Press, Inc.

United States

Publication History

Published: 01 July 2022

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Cited By

Cambie SCames van Batenburg WCranston D(2024)Optimally reconfiguring list and correspondence colouringsEuropean Journal of Combinatorics10.1016/j.ejc.2023.103798115:COnline publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1016/j.ejc.2023.103798

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