32 research outputs found
Coloring Graphs having Few Colorings over Path Decompositions
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no
time algorithm for
deciding if an -vertex graph with pathwidth
admits a proper vertex coloring with colors unless the Strong Exponential
Time Hypothesis (SETH) is false. We show here that nevertheless, when
, where is the maximum degree in the
graph , there is a better algorithm, at least when there are few colorings.
We present a Monte Carlo algorithm that given a graph along with a path
decomposition of with pathwidth runs in time, that
distinguishes between -colorable graphs having at most proper
-colorings and non--colorable graphs. We also show how to obtain a
-coloring in the same asymptotic running time. Our algorithm avoids
violating SETH for one since high degree vertices still cost too much and the
mentioned hardness construction uses a lot of them.
We exploit a new variation of the famous Alon--Tarsi theorem that has an
algorithmic advantage over the original form. The original theorem shows a
graph has an orientation with outdegree less than at every vertex, with a
different number of odd and even Eulerian subgraphs only if the graph is
-colorable, but there is no known way of efficiently finding such an
orientation. Our new form shows that if we instead count another difference of
even and odd subgraphs meeting modular degree constraints at every vertex
picked uniformly at random, we have a fair chance of getting a non-zero value
if the graph has few -colorings. Yet every non--colorable graph gives a
zero difference, so a random set of constraints stands a good chance of being
useful for separating the two cases.Comment: Strengthened result from uniquely -colorable graphs to graphs with
few -colorings. Also improved running tim
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