12,140 research outputs found

    Coloring permutation-gain graphs

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    Correspondence colorings of graphs were introduced in 2018 by Dvořák and Postle as a generalization of list colorings of graphs which generalizes ordinary graph coloring. Kim and Ozeki observed that correspondence colorings generalize various notions of signed-graph colorings which again generalizes ordinary graph colorings. In this note we state how correspondence colorings generalize Zaslavsky's notion of gain-graph colorings and then formulate a new coloring theory of permutation-gain graphs that sits between gain-graph coloring and correspondence colorings. Like Zaslavsky's gain-graph coloring, our new notion of coloring permutation-gain graphs has well defined chromatic polynomials and lifts to colorings of the regular covering graph of a permutation-gain graph

    Recoloring bounded treewidth graphs

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    Let kk be an integer. Two vertex kk-colorings of a graph are \emph{adjacent} if they differ on exactly one vertex. A graph is \emph{kk-mixing} if any proper kk-coloring can be transformed into any other through a sequence of adjacent proper kk-colorings. Any graph is (tw+2)(tw+2)-mixing, where twtw is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two (tw+2)(tw+2)-colorings is at most quadratic, a problem left open in Bonamy et al. (2012). Jerrum proved that any graph is kk-mixing if kk is at least the maximum degree plus two. We improve Jerrum's bound using the grundy number, which is the worst number of colors in a greedy coloring.Comment: 11 pages, 5 figure
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