608 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
Pushable chromatic number of graphs with degree constraints
International audiencePushable homomorphisms and the pushable chromatic number of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph , we have , where denotes the oriented chromatic number of . This stands as the first general bounds on . This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all , we first prove that the maximum value of the pushable chromatic number of a connected oriented graph with maximum degree lies between and which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when , we then prove that the maximum value of the pushable chromatic number is~ or~. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~ lies between~ and~. The former upper bound of~ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Novel Split-Based Approaches to Computing Phylogenetic Diversity and Planar Split Networks
EThOS - Electronic Theses Online ServiceGBUnited Kingdo
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