1,805 research outputs found
Locally identifying coloring in bounded expansion classes of graphs
A proper vertex coloring of a graph is said to be locally identifying if the
sets of colors in the closed neighborhood of any two adjacent non-twin vertices
are distinct. The lid-chromatic number of a graph is the minimum number of
colors used by a locally identifying vertex-coloring. In this paper, we prove
that for any graph class of bounded expansion, the lid-chromatic number is
bounded. Classes of bounded expansion include minor closed classes of graphs.
For these latter classes, we give an alternative proof to show that the
lid-chromatic number is bounded. This leads to an explicit upper bound for the
lid-chromatic number of planar graphs. This answers in a positive way a
question of Esperet et al [L. Esperet, S. Gravier, M. Montassier, P. Ochem and
A. Parreau. Locally identifying coloring of graphs. Electronic Journal of
Combinatorics, 19(2), 2012.]
Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups
We study in this paper some connections between the Fraisse theory of
amalgamation classes and ultrahomogeneous structures, Ramsey theory, and
topological dynamics of automorphism groups of countable structures.Comment: 73 pages, LaTeX 2e, to appear in Geom. Funct. Ana
Kernelization and Sparseness: the case of Dominating Set
We prove that for every positive integer and for every graph class
of bounded expansion, the -Dominating Set problem admits a
linear kernel on graphs from . Moreover, when is only
assumed to be nowhere dense, then we give an almost linear kernel on for the classic Dominating Set problem, i.e., for the case . These
results generalize a line of previous research on finding linear kernels for
Dominating Set and -Dominating Set. However, the approach taken in this
work, which is based on the theory of sparse graphs, is radically different and
conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related Connected
Dominating Set problem, the existence of such kernelization algorithms is
unlikely, even though the problem is known to admit a linear kernel on
-topological-minor-free graphs. Also, we prove that for any somewhere dense
class , there is some for which -Dominating Set is
W[]-hard on . Thus, our results fall short of proving a sharp
dichotomy for the parameterized complexity of -Dominating Set on
subgraph-monotone graph classes: we conjecture that the border of tractability
lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded
expansion graph
On Locally Identifying Coloring of Cartesian Product and Tensor Product of Graphs
For a positive integer , a proper -coloring of a graph is a mapping
such that for each
edge . The smallest integer for which there is a proper
-coloring of is called chromatic number of , denoted by .
A \emph{locally identifying coloring} (for short, lid-coloring) of a graph
is a proper -coloring of such that every pair of adjacent vertices
with distinct closed neighborhoods has distinct set of colors in their closed
neighborhoods.
The smallest integer such that has a lid-coloring with colors is
called
\emph{locally identifying chromatic number}
(for short, \emph{lid-chromatic number}) of , denoted by .
In this paper, we study lid-coloring of Cartesian product and tensor product
of two graphs. We prove that if and are two connected graphs having at
least two vertices then (a)
and (b) . Here and
denote the Cartesian and tensor products of and
respectively. We also give exact values of lid-chromatic number of Cartesian
product (resp. tensor product) of two paths, a cycle and a path, and two
cycles
- …