31 research outputs found
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
Choosability of the square of planar subcubic graphs with large girth
We first show that the choose number of the square of a subcubic graph with maximum average degree less than 18/7 is at most 6. As a corollary, we get that the choose number of the square of a planar graph with girth at least 9 is at most 6. We then show that the choose number of the square of a subcubic planar graph with girth at least 13 is at most 5
Linear colorings of subcubic graphs
A linear coloring of a graph is a proper coloring of the vertices of the
graph so that each pair of color classes induce a union of disjoint paths. In
this paper, we prove that for every connected graph with maximum degree at most
three and every assignment of lists of size four to the vertices of the graph,
there exists a linear coloring such that the color of each vertex belongs to
the list assigned to that vertex and the neighbors of every degree-two vertex
receive different colors, unless the graph is or . This confirms
a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is
constructive and yields a linear-time algorithm to find such a coloring
Choosability of a weighted path and free-choosability of a cycle
A graph with a list of colors and weight for each vertex
is -colorable if one can choose a subset of colors from
for each vertex , such that adjacent vertices receive disjoint color
sets. In this paper, we give necessary and sufficient conditions for a weighted
path to be -colorable for some list assignments . Furthermore, we
solve the problem of the free-choosability of a cycle.Comment: 9 page