List recoloring of planar graphs
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A list assignment L of a graph G is a function that assigns to every vertex v of G a set L(v) of colors. A proper coloring α of G is called an L-coloring of G if α(v)∈ L(v) for every v∈ V(G). For a list assignment L of G, the L-recoloring graph 𝒢(G,L) of G is a graph whose vertices correspond to the L-colorings of G and two vertices of 𝒢(G,L) are adjacent if their corresponding L-colorings differ at exactly one vertex of G. A d-face in a plane graph is a face of length d. Dvořák and Feghali conjectured for a planar graph G and a list assignment L of G, that: (i) If |L(v)|≥ 10 for every v∈ V(G), then the diameter of 𝒢(G,L) is O(|V(G)|). (ii) If G is triangle-free and |L(v)|≥ 7 for every v∈ V(G), then the diameter of 𝒢(G,L) is O(|V(G)|). In a recent paper, Cranston (European J. Combin. (2022)) has proved (ii). In this paper, we prove the following results. Let G be a plane graph and L be a list assignment of G. ∙ If for every 3-face of G, there are at most two 3-faces adjacent to it and |L(v)|≥ 10 for every v∈ V(G), then the diameter of 𝒢(G,L) is at most 190|V(G)|. ∙ If for every 3-face of G, there is at most one 3-face adjacent to it and |L(v)|≥ 9 for every v∈ V(G), then the diameter of 𝒢(G,L) is at most 13|V(G)|. ∙ If the faces adjacent to any 3-face have length at least 6 and |L(v)|≥ 7 for every v∈ V(G), then the diameter of 𝒢(G,L) is at most 242|V(G)|. This result strengthens the Cranston's result on (ii).
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