2 research outputs found
Coloring squares of graphs with mad constraints
A proper vertex -coloring of a graph is an assignment of colors to the vertices of the graph such that no two
adjacent vertices are associated with the same color. The square of a
graph is the graph defined by and if and only
if the distance between and is at most two. We denote by
the chromatic number of , which is the least integer such that a
-coloring of exists. By definition, at least colors are
needed for this goal, where denotes the maximum degree of the graph
. In this paper, we prove that the square of every graph with
and is -choosable and
even correspondence-colorable. Furthermore, we show a family of -degenerate
graphs with , arbitrarily large maximum degree, and
, improving the result of Kim and
Park.Comment: 14 pages, 4 figure
Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)
We survey work on coloring, list coloring, and painting squares of graphs; in
particular, we consider strong edge-coloring. We focus primarily on planar
graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography,
comments are welcome, published as a Dynamic Survey in Electronic Journal of
Combinatoric