1,011 research outputs found
Asymmetric coloring games on incomparability graphs
Consider the following game on a graph : Alice and Bob take turns coloring
the vertices of properly from a fixed set of colors; Alice wins when the
entire graph has been colored, while Bob wins when some uncolored vertices have
been left. The game chromatic number of is the minimum number of colors
that allows Alice to win the game. The game Grundy number of is defined
similarly except that the players color the vertices according to the first-fit
rule and they only decide on the order in which it is applied. The -game
chromatic and Grundy numbers are defined likewise except that Alice colors
vertices and Bob colors vertices in each round. We study the behavior of
these parameters for incomparability graphs of posets with bounded width. We
conjecture a complete characterization of the pairs for which the
-game chromatic and Grundy numbers are bounded in terms of the width of
the poset; we prove that it gives a necessary condition and provide some
evidence for its sufficiency. We also show that the game chromatic number is
not bounded in terms of the Grundy number, which answers a question of Havet
and Zhu
Acyclic edge-coloring using entropy compression
An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G
and every cycle contains at least three colors. We prove that every graph with
maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4
colors, improving the previous bound of 9.62 (Delta - 1). Our bound results
from the analysis of a very simple randomised procedure using the so-called
entropy compression method. We show that the expected running time of the
procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices
and edges of G. Such a randomised procedure running in expected polynomial time
was only known to exist in the case where at least 16 Delta colors were
available. Our aim here is to make a pedagogic tutorial on how to use these
ideas to analyse a broad range of graph coloring problems. As an application,
also show that every graph with maximum degree Delta has a star coloring with 2
sqrt(2) Delta^{3/2} + Delta colors.Comment: 13 pages, revised versio
Structural properties of 1-planar graphs and an application to acyclic edge coloring
A graph is called 1-planar if it can be drawn on the plane so that each edge
is crossed by at most one other edge. In this paper, we establish a local
property of 1-planar graphs which describes the structure in the neighborhood
of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some
new classes of light graphs in 1-planar graphs with the bounded degree are
found. Therefore, two open problems presented by Fabrici and Madaras [The
structure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] are
solved. Furthermore, we prove that each 1-planar graph with maximum degree
is acyclically edge -choosable where
.Comment: Please cite this published article as: X. Zhang, G. Liu, J.-L. Wu.
Structural properties of 1-planar graphs and an application to acyclic edge
coloring. Scientia Sinica Mathematica, 2010, 40, 1025--103
Local Conflict Coloring
Locally finding a solution to symmetry-breaking tasks such as
vertex-coloring, edge-coloring, maximal matching, maximal independent set,
etc., is a long-standing challenge in distributed network computing. More
recently, it has also become a challenge in the framework of centralized local
computation. We introduce conflict coloring as a general symmetry-breaking task
that includes all the aforementioned tasks as specific instantiations ---
conflict coloring includes all locally checkable labeling tasks from
[Naor\&Stockmeyer, STOC 1993]. Conflict coloring is characterized by two
parameters and , where the former measures the amount of freedom given
to the nodes for selecting their colors, and the latter measures the number of
constraints which colors of adjacent nodes are subject to.We show that, in the
standard LOCAL model for distributed network computing, if l/d \textgreater{}
\Delta, then conflict coloring can be solved in rounds in -node graphs with maximum degree
, where ignores the polylog factors in . The
dependency in~ is optimal, as a consequence of the lower
bound by [Linial, SIAM J. Comp. 1992] for -coloring. An important
special case of our result is a significant improvement over the best known
algorithm for distributed -coloring due to [Barenboim, PODC 2015],
which required rounds. Improvements for other
variants of coloring, including -list-coloring,
-edge-coloring, -coloring, etc., also follow from our general
result on conflict coloring. Likewise, in the framework of centralized local
computation algorithms (LCAs), our general result yields an LCA which requires
a smaller number of probes than the previously best known algorithm for
vertex-coloring, and works for a wide range of coloring problems
Intersection representation of digraphs in trees with few leaves
The leafage of a digraph is the minimum number of leaves in a host tree in
which it has a subtree intersection representation. We discuss bounds on the
leafage in terms of other parameters (including Ferrers dimension), obtaining a
string of sharp inequalities.Comment: 12 pages, 3 included figure
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