32 research outputs found
On lower bounds for the matching number of subcubic graphs
We give a complete description of the set of triples (a,b,c) of real numbers
with the following property. There exists a constant K such that a n_3 + b n_2
+ c n_1 - K is a lower bound for the matching number of every connected
subcubic graph G, where n_i denotes the number of vertices of degree i for each
i
Goldberg's Conjecture is true for random multigraphs
In the 70s, Goldberg, and independently Seymour, conjectured that for any
multigraph , the chromatic index satisfies , where . We show that their conjecture (in a
stronger form) is true for random multigraphs. Let be the probability
space consisting of all loopless multigraphs with vertices and edges,
in which pairs from are chosen independently at random with
repetitions. Our result states that, for a given ,
typically satisfies . In
particular, we show that if is even and , then
for a typical . Furthermore, for a fixed
, if is odd, then a typical has
for , and
for .Comment: 26 page
Morphing Planar Graph Drawings with Unidirectional Moves
Alamdari et al. showed that given two straight-line planar drawings of a
graph, there is a morph between them that preserves planarity and consists of a
polynomial number of steps where each step is a \emph{linear morph} that moves
each vertex at constant speed along a straight line. An important step in their
proof consists of converting a \emph{pseudo-morph} (in which contractions are
allowed) to a true morph. Here we introduce the notion of \emph{unidirectional
morphing} step, where the vertices move along lines that all have the same
direction. Our main result is to show that any planarity preserving
pseudo-morph consisting of unidirectional steps and contraction of low degree
vertices can be turned into a true morph without increasing the number of
steps. Using this, we strengthen Alamdari et al.'s result to use only
unidirectional morphs, and in the process we simplify the proof.Comment: 13 pages, 9 figure
A note on the Size-Ramsey number of long subdivisions of graphs
Let TsH be the graph obtained from a given graph H by subdividing each
edge s times. Motivated by a problem raised by Igor Pak [Mixing
time and long paths in graphs, in Proc.âof the 13th annual ACM-SIAM
Symposium on Discrete Algorithms (SODA 2002) 321â328], we prove
that, for any graph H, there exist graphs G with O(s) edges that are
Ramsey with respect to TsH
The Ramsey Number for 3-Uniform Tight Hypergraph Cycles
Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .â.â., vn and edges v1v2v3, v2v3v4, .â.â., vnâ1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every redâblue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and RĂśdl