1,480 research outputs found
Distance colouring without one cycle length
We consider distance colourings in graphs of maximum degree at most and
how excluding one fixed cycle length affects the number of colours
required as . For vertex-colouring and , if any two
distinct vertices connected by a path of at most edges are required to be
coloured differently, then a reduction by a logarithmic (in ) factor against
the trivial bound can be obtained by excluding an odd cycle length
if is odd or by excluding an even cycle length . For edge-colouring and , if any two distinct edges connected by
a path of fewer than edges are required to be coloured differently, then
excluding an even cycle length is sufficient for a logarithmic
factor reduction. For , neither of the above statements are possible
for other parity combinations of and . These results can be
considered extensions of results due to Johansson (1996) and Mahdian (2000),
and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang
(2014).Comment: 14 pages, 1 figur
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Maximum -edge-colorable subgraphs of class II graphs
A graph is class II, if its chromatic index is at least . Let
be a maximum -edge-colorable subgraph of . The paper proves best
possible lower bounds for , and structural properties of
maximum -edge-colorable subgraphs. It is shown that every set of
vertex-disjoint cycles of a class II graph with can be extended
to a maximum -edge-colorable subgraph. Simple graphs have a maximum
-edge-colorable subgraph such that the complement is a matching.
Furthermore, a maximum -edge-colorable subgraph of a simple graph is
always class I.Comment: 13 pages, 2 figures, the proof of the Lemma 1 is correcte
Bipartite induced density in triangle-free graphs
We prove that any triangle-free graph on vertices with minimum degree at
least contains a bipartite induced subgraph of minimum degree at least
. This is sharp up to a logarithmic factor in . Relatedly, we show
that the fractional chromatic number of any such triangle-free graph is at most
the minimum of and as . This is
sharp up to constant factors. Similarly, we show that the list chromatic number
of any such triangle-free graph is at most as
.
Relatedly, we also make two conjectures. First, any triangle-free graph on
vertices has fractional chromatic number at most
as . Second, any triangle-free
graph on vertices has list chromatic number at most as
.Comment: 20 pages; in v2 added note of concurrent work and one reference; in
v3 added more notes of ensuing work and a result towards one of the
conjectures (for list colouring
Edge-coloring via fixable subgraphs
Many graph coloring proofs proceed by showing that a minimal counterexample
to the theorem being proved cannot contain certain configurations, and then
showing that each graph under consideration contains at least one such
configuration; these configurations are called \emph{reducible} for that
theorem. (A \emph{configuration} is a subgraph , along with specified
degrees in the original graph for each vertex of .)
We give a general framework for showing that configurations are reducible for
edge-coloring. A particular form of reducibility, called \emph{fixability}, can
be considered without reference to a containing graph. This has two key
benefits: (i) we can now formulate necessary conditions for fixability, and
(ii) the problem of fixability is easy for a computer to solve. The necessary
condition of \emph{superabundance} is sufficient for multistars and we
conjecture that it is sufficient for trees as well, which would generalize the
powerful technique of Tashkinov trees.
Via computer, we can generate thousands of reducible configurations, but we
have short proofs for only a small fraction of these. The computer can write
\LaTeX\ code for its proofs, but they are only marginally enlightening and can
run thousands of pages long. We give examples of how to use some of these
reducible configurations to prove conjectures on edge-coloring for small
maximum degree. Our aims in writing this paper are (i) to provide a common
context for a variety of reducible configurations for edge-coloring and (ii) to
spur development of methods for humans to understand what the computer already
knows.Comment: 18 pages, 8 figures; 12-page appendix with 39 figure
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
- …