Recolouring weakly chordal graphs and the complement of triangle-free graphs

Merkel, Owen

doi:10.1016/j.disc.2021.112708

Cited by 3 publications

(3 citation statements)

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“…Biedl, Lubiw, and Merkel [1] showed that every P 4 -sparse graph is ℓ-mixing and the ℓrecolouring diameter is at most 4n 2 . The last author [11] showed that for all p ≥ 1, there exists a k-colourable weakly chordal graph that is not (k + p)-mixing.…”

Section: Theorem 2 ([8]mentioning

confidence: 99%

“…Theorem 10 ( [11]). Every 3K 1 -free graph is ℓ-mixing and the ℓ-recolouring diameter is at most 4n.…”

Section: Frozen Colouringsmentioning

confidence: 99%

“…Question 1 has been answered completely when H is a 3-vertex graph, namely 3K 1 , P 2 + P 1 , P 3 , and K 3 . The last author [11] showed that every 3K 1 -free graph is ℓ-mixing and the ℓrecolouring diameter is at most 4n. The class of (P 2 + P 1 )-free graphs is a subclass of P 4 -free graphs, and so we refer to the results on P 4 -free graphs in this case (see Theorem 1).…”

Section: Introductionmentioning

confidence: 99%

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Reconfiguration of vertex colouring and forbidden induced subgraphs

Belavadi¹,

Cameron²,

Merkel³

2022

Preprint

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The reconfiguration graph of the k-colourings, denoted R k (G), is the graph whose vertices are the k-colourings of G and two colourings are adjacent in R k (G) if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of R k+1 (G) for a k-colourable graph G restricted by forbidden induced subgraphs. We show that R k+1 (G) is connected for every k-colourable H-free graph G if and only if H is an induced subgraph of P 4 or P 3 + P 1 . We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if G is a k-colourable (2K 2 , C 4 )-free graph, then R k+1 (G) is connected with diameter at most 4n. Furthermore, we show that R k+1 (G) is connected for every k-colourable (P 5 , C 4 )-free graph G.

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Section: Theorem 2 ([8]mentioning

confidence: 99%

“…Theorem 10 ( [11]). Every 3K 1 -free graph is ℓ-mixing and the ℓ-recolouring diameter is at most 4n.…”

Section: Frozen Colouringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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