353 research outputs found
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
Distance-two coloring of sparse graphs
Consider a graph and, for each vertex , a subset
of neighbors of . A -coloring is a coloring of the
elements of so that vertices appearing together in some receive
pairwise distinct colors. An obvious lower bound for the minimum number of
colors in such a coloring is the maximum size of a set , denoted by
. In this paper we study graph classes for which there is a
function , such that for any graph and any , there is a
-coloring using at most colors. It is proved that if
such a function exists for a class , then can be taken to be a linear
function. It is also shown that such classes are precisely the classes having
bounded star chromatic number. We also investigate the list version and the
clique version of this problem, and relate the existence of functions bounding
those parameters to the recently introduced concepts of classes of bounded
expansion and nowhere-dense classes.Comment: 13 pages - revised versio
Boxicity of graphs on surfaces
The boxicity of a graph is the least integer for which there
exist interval graphs , , such that . Scheinerman proved in 1984 that outerplanar graphs have boxicity
at most two and Thomassen proved in 1986 that planar graphs have boxicity at
most three. In this note we prove that the boxicity of toroidal graphs is at
most 7, and that the boxicity of graphs embeddable in a surface of
genus is at most . This result yields improved bounds on the
dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure
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