12,140 research outputs found
Coloring permutation-gain graphs
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
Let be an integer. Two vertex -colorings of a graph are
\emph{adjacent} if they differ on exactly one vertex. A graph is
\emph{-mixing} if any proper -coloring can be transformed into any other
through a sequence of adjacent proper -colorings. Any graph is
-mixing, where is the treewidth of the graph (Cereceda 2006). We
prove that the shortest sequence between any two -colorings is at most
quadratic, a problem left open in Bonamy et al. (2012).
Jerrum proved that any graph is -mixing if 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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