101 research outputs found
Fractional colorings of cubic graphs with large girth
We show that every (sub)cubic n-vertex graph with sufficiently large girth
has fractional chromatic number at most 2.2978 which implies that it contains
an independent set of size at least 0.4352n. Our bound on the independence
number is valid to random cubic graphs as well as it improves existing lower
bounds on the maximum cut in cubic graphs with large girth
Fractional colorings of cubic graphs with large girth
International audienceWe show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978, which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid for random cubic graphs as well, as it improves existing lower bounds on the maximum cut in cubic graphs with large girth
On Colorings of Graph Powers
In this paper, some results concerning the colorings of graph powers are
presented. The notion of helical graphs is introduced. We show that such graphs
are hom-universal with respect to high odd-girth graphs whose st power
is bounded by a Kneser graph. Also, we consider the problem of existence of
homomorphism to odd cycles. We prove that such homomorphism to a -cycle
exists if and only if the chromatic number of the st power of
is less than or equal to 3, where is the 2-subdivision of . We also
consider Ne\v{s}et\v{r}il's Pentagon problem. This problem is about the
existence of high girth cubic graphs which are not homomorphic to the cycle of
size five. Several problems which are closely related to Ne\v{s}et\v{r}il's
problem are introduced and their relations are presented
Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs
We show that for every cubic graph G with sufficiently large girth there
exists a probability distribution on edge-cuts of G such that each edge is in a
randomly chosen cut with probability at least 0.88672. This implies that G
contains an edge-cut of size at least 1.33008n, where n is the number of
vertices of G, and has fractional cut covering number at most 1.127752. The
lower bound on the size of maximum edge-cut also applies to random cubic
graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge cut
of size 1.33008n
Generalized Colorings of Graphs
A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique
Disjoint list-colorings for planar graphs
One of Thomassen's classical results is that every planar graph of girth at
least is 3-choosable. One can wonder if for a planar graph of girth
sufficiently large and a -list-assignment , one can do even better. Can
one find disjoint -colorings (a packing), or disjoint -colorings,
or a collection of -colorings that to every vertex assigns every color on
average in one third of the cases (a fractional packing)? We prove that the
packing is impossible, but two disjoint -colorings are guaranteed if the
girth is at least , and a fractional packing exists when the girth is at
least
For a graph , the least such that there are always disjoint proper
list-colorings whenever we have lists all of size associated to the
vertices is called the list packing number of . We lower the
two-times-degeneracy upper bound for the list packing number of planar graphs
of girth or . As immediate corollaries, we improve bounds for
-flexibility of classes of planar graphs with a given girth. For
instance, where previously Dvo\v{r}\'{a}k et al. proved that planar graphs of
girth are (weighted) -flexibly -choosable for an extremely
small value of , we obtain the optimal value .
Finally, we completely determine and show interesting behavior on the packing
numbers for -minor-free graphs for some small graphs Comment: 36 pages, 8 figure
Independent sets and cuts in large-girth regular graphs
We present a local algorithm producing an independent set of expected size
on large-girth 3-regular graphs and on large-girth
4-regular graphs. We also construct a cut (or bisection or bipartite subgraph)
with edges on large-girth 3-regular graphs. These decrease the gaps
between the best known upper and lower bounds from to , from
to and from to , respectively. We are using
local algorithms, therefore, the method also provides upper bounds for the
fractional coloring numbers of and and fractional edge coloring number . Our algorithms are applications of the technique introduced by Hoppen
and Wormald
Circular edge-colorings of cubic graphs with girth six
We show that the circular chromatic index of a (sub)cubic graph with girth at
least six is at most 7/2.Comment: 13 pages, 6 figure
Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)
We survey work on coloring, list coloring, and painting squares of graphs; in
particular, we consider strong edge-coloring. We focus primarily on planar
graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography,
comments are welcome, published as a Dynamic Survey in Electronic Journal of
Combinatoric
Spotting Trees with Few Leaves
We show two results related to the Hamiltonicity and -Path algorithms in
undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10].
First, we demonstrate that the technique used can be generalized to finding
some -vertex tree with leaves in an -vertex undirected graph in
time. It can be applied as a subroutine to solve the
-Internal Spanning Tree (-IST) problem in
time using polynomial space, improving upon previous algorithms for this
problem. In particular, for the first time we break the natural barrier of
. Second, we show that the iterated random bipartition employed by
the algorithm can be improved whenever the host graph admits a vertex coloring
with few colors; it can be an ordinary proper vertex coloring, a fractional
vertex coloring, or a vector coloring. In effect, we show improved bounds for
-Path and Hamiltonicity in any graph of maximum degree
or with vector chromatic number at most 8
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