2,289 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
Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies
We settle a problem of Havel by showing that there exists an absolute
constant d such that if G is a planar graph in which every two distinct
triangles are at distance at least d, then G is 3-colorable. In fact, we prove
a more general theorem. Let G be a planar graph, and let H be a set of
connected subgraphs of G, each of bounded size, such that every two distinct
members of H are at least a specified distance apart and all triangles of G are
contained in \bigcup{H}. We give a sufficient condition for the existence of a
3-coloring phi of G such that for every B\in H, the restriction of phi to B is
constrained in a specified way.Comment: 26 pages, no figures. Updated presentatio
Compact Labelings For Efficient First-Order Model-Checking
We consider graph properties that can be checked from labels, i.e., bit
sequences, of logarithmic length attached to vertices. We prove that there
exists such a labeling for checking a first-order formula with free set
variables in the graphs of every class that is \emph{nicely locally
cwd-decomposable}. This notion generalizes that of a \emph{nicely locally
tree-decomposable} class. The graphs of such classes can be covered by graphs
of bounded \emph{clique-width} with limited overlaps. We also consider such
labelings for \emph{bounded} first-order formulas on graph classes of
\emph{bounded expansion}. Some of these results are extended to counting
queries
- …