5,265 research outputs found
On star edge colorings of bipartite and subcubic graphs
A star edge coloring of a graph is a proper edge coloring with no -colored
path or cycle of length four. The star chromatic index of
is the minimum number for which has a star edge coloring with
colors. We prove upper bounds for the star chromatic index of complete
bipartite graphs; in particular we obtain tight upper bounds for the case when
one part has size at most . We also consider bipartite graphs where all
vertices in one part have maximum degree and all vertices in the other part
has maximum degree . Let be an integer (), we prove that if
then ; and if , then ; both upper bounds are sharp.
Finally, we consider the well-known conjecture that subcubic graphs have star
chromatic index at most ; in particular we settle this conjecture for cubic
Halin graphs.Comment: 18 page
Vertex-Coloring with Star-Defects
Defective coloring is a variant of traditional vertex-coloring, according to
which adjacent vertices are allowed to have the same color, as long as the
monochromatic components induced by the corresponding edges have a certain
structure. Due to its important applications, as for example in the
bipartisation of graphs, this type of coloring has been extensively studied,
mainly with respect to the size, degree, and acyclicity of the monochromatic
components.
In this paper we focus on defective colorings in which the monochromatic
components are acyclic and have small diameter, namely, they form stars. For
outerplanar graphs, we give a linear-time algorithm to decide if such a
defective coloring exists with two colors and, in the positive case, to
construct one. Also, we prove that an outerpath (i.e., an outerplanar graph
whose weak-dual is a path) always admits such a two-coloring. Finally, we
present NP-completeness results for non-planar and planar graphs of bounded
degree for the cases of two and three colors
Star 5-edge-colorings of subcubic multigraphs
The star chromatic index of a multigraph , denoted , is the
minimum number of colors needed to properly color the edges of such that no
path or cycle of length four is bi-colored. A multigraph is star
-edge-colorable if . Dvo\v{r}\'ak, Mohar and \v{S}\'amal
[Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every
subcubic multigraph is star -edge-colorable, and conjectured that every
subcubic multigraph should be star -edge-colorable. Kerdjoudj, Kostochka and
Raspaud considered the list version of this problem for simple graphs and
proved that every subcubic graph with maximum average degree less than is
star list--edge-colorable. It is known that a graph with maximum average
degree is not necessarily star -edge-colorable. In this paper, we
prove that every subcubic multigraph with maximum average degree less than
is star -edge-colorable.Comment: to appear in Discrete Mathematics. arXiv admin note: text overlap
with arXiv:1701.0410
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
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
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