114 research outputs found
A Unified Approach to Distance-Two Colouring of Graphs on Surfaces
In this paper we introduce the notion of -colouring of a graph :
For given subsets of neighbours of , for every , this
is a proper colouring of the vertices of such that, in addition, vertices
that appear together in some receive different colours. This
concept generalises the notion of colouring the square of graphs and of cyclic
colouring of graphs embedded in a surface. We prove a general result for graphs
embeddable in a fixed surface, which implies asymptotic versions of Wegner's
and Borodin's Conjecture on the planar version of these two colourings. Using a
recent approach of Havet et al., we reduce the problem to edge-colouring of
multigraphs, and then use Kahn's result that the list chromatic index is close
to the fractional chromatic index.
Our results are based on a strong structural lemma for graphs embeddable in a
fixed surface, which also implies that the size of a clique in the square of a
graph of maximum degree embeddable in some fixed surface is at most
plus a constant.Comment: 36 page
Coloring decompositions of complete geometric graphs
A decomposition of a non-empty simple graph is a pair , such that
is a set of non-empty induced subgraphs of , and every edge of
belongs to exactly one subgraph in . The chromatic index of a
decomposition is the smallest number for which there exists a
-coloring of the elements of in such a way that: for every element of
all of its edges have the same color, and if two members of share at
least one vertex, then they have different colors. A long standing conjecture
of Erd\H{o}s-Faber-Lov\'asz states that every decomposition of the
complete graph satisfies . In this paper we work
with geometric graphs, and inspired by this formulation of the conjecture, we
introduce the concept of chromatic index of a decomposition of the complete
geometric graph. We present bounds for the chromatic index of several types of
decompositions when the vertices of the graph are in general position. We also
consider the particular case in which the vertices are in convex position and
present bounds for the chromatic index of a few types of decompositions.Comment: 18 pages, 5 figure
Some results on (a:b)-choosability
A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing
that if a graph is -choosable, and , then is not
necessarily -choosable. Applying probabilistic methods, an upper bound
for the choice number of a graph is given. We also prove that a
directed graph with maximum outdegree and no odd directed cycle is
-choosable for every . Other results presented in this
article are related to the strong choice number of graphs (a generalization of
the strong chromatic number). We conclude with complexity analysis of some
decision problems related to graph choosability
Generation and Properties of Snarks
For many of the unsolved problems concerning cycles and matchings in graphs
it is known that it is sufficient to prove them for \emph{snarks}, the class of
nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part
of this paper we present a new algorithm for generating all non-isomorphic
snarks of a given order. Our implementation of the new algorithm is 14 times
faster than previous programs for generating snarks, and 29 times faster for
generating weak snarks. Using this program we have generated all non-isomorphic
snarks on vertices. Previously lists up to vertices have been
published. In the second part of the paper we analyze the sets of generated
snarks with respect to a number of properties and conjectures. We find that
some of the strongest versions of the cycle double cover conjecture hold for
all snarks of these orders, as does Jaeger's Petersen colouring conjecture,
which in turn implies that Fulkerson's conjecture has no small counterexamples.
In contrast to these positive results we also find counterexamples to eight
previously published conjectures concerning cycle coverings and the general
cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated
and typos corrected. This version differs from the published one in that the
Arxiv-version has data about the automorphisms of snarks; Journal of
Combinatorial Theory. Series B. 201
The strong chromatic index of 1-planar graphs
The chromatic index of a graph is the smallest for which
admits an edge -coloring such that any two adjacent edges have distinct
colors. The strong chromatic index of is the smallest such
that has a proper edge -coloring with the condition that any two edges
at distance at most 2 receive distinct colors. A graph is 1-planar if it can be
drawn in the plane so that each edge is crossed by at most one other edge.
In this paper, we show that every graph with maximum average degree
has . As a corollary, we
prove that every 1-planar graph with maximum degree has
, which improves a result, due to Bensmail et
al., which says that if
Some snarks are worse than others
Many conjectures and open problems in graph theory can either be reduced to
cubic graphs or are directly stated for cubic graphs. Furthermore, it is known
that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless
cubic graph which is not 3--edge-colourable. In this paper we deal with the
fact that the family of potential counterexamples to many interesting
conjectures can be narrowed even further to the family of
bridgeless cubic graphs whose edge set cannot be covered with four perfect
matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover
Conjecture and the Fan-Raspaud Conjecture are examples of statements for which
is crucial. In this paper, we study parameters which have
the potential to further refine and thus enlarge the set of
cubic graphs for which the mentioned conjectures can be verified. We show that
can be naturally decomposed into subsets with increasing
complexity, thereby producing a natural scale for proving these conjectures.
More precisely, we consider the following parameters and questions: given a
bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii)
how many copies of the same perfect matching need to be added, and (iii) how
many 2--factors need to be added so that the resulting regular graph is Class
I? We present new results for these parameters and we also establish some
strong relations between these problems and some long-standing conjectures.Comment: 27 pages, 16 figure
Interval Edge-Colorings of Graphs
A proper edge-coloring of a graph G by positive integers is called an interval edge-coloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edge-colorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edge-colorings to schedule parent-teacher conferences so that every person\u27s conferences occur in consecutive slots. A solution exists if and only if the bipartite graph with vertices for parents and teachers, and edges for the required meetings, has an interval edge-coloring. A well-known result of Vizing states that for any simple graph G, χ0(G) ≤ ∆(G)+1, where χ0(G) and ∆(G) denote the edge-chromatic number and maximum degree of G, respectively. A graph G is called class 1 if χ0(G) = ∆(G), and class 2 if χ0(G) = ∆(G) + 1. One can see that any graph admitting an interval edge-coloring must be of class 1, and thus every graph of class 2 does not have such a coloring. Finding an interval edge-coloring of a given graph is hard. In fact, it has been shown that determining whether a bipartite graph has an interval edge-coloring is NP-complete. In this thesis, we survey known results on interval edge-colorings of graphs, with a focus on the progress of (a, b)-biregular bipartite graphs. Discussion of related topics and future work is included at the end. We also give a new proof of Theorem 3.15 on the existence of proper path factors of (3, 4)-biregular graphs. Finally, we obtain a new result, Theorem 3.18, which states that if a proper path factor of any (3, 4)-biregular graph has no path of length 8, then it contains paths of length 6 only. The new result we obtained and the method we developed in the proof of Theorem 3.15 might be helpful in attacking the open problems mentioned in the Future Work section of Chapter 5
- …