6,384 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.]
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
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
Coarse distinguishability of graphs with symmetric growth
Let be a connected, locally finite graph with symmetric growth. We prove
that there is a vertex coloring and some
such that every automorphism preserving is
-close to the identity map; this can be seen as a coarse geometric version
of symmetry breaking. We also prove that the infinite motion conjecture is true
for graphs where at least one vertex stabilizer satisfies the following
condition: for every non-identity automorphism , there is a sequence
such that
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