20 research outputs found
Progress on the adjacent vertex distinguishing edge colouring conjecture
A proper edge colouring of a graph is adjacent vertex distinguishing if no
two adjacent vertices see the same set of colours. Using a clever application
of the Local Lemma, Hatami (2005) proved that every graph with maximum degree
and no isolated edge has an adjacent vertex distinguishing edge
colouring with colours, provided is large enough. We
show that this bound can be reduced to . This is motivated by the
conjecture of Zhang, Liu, and Wang (2002) that colours are enough
for .Comment: v2: Revised following referees' comment
Planar graphs with maximum degree Delta\geq 9 are (\Delta+1)-edge-choosable -- short proof
We give a short proof of the following theorem due to Borodin~\cite{Bor90}. Every planar graph with maximum degree is -edge-choosable
Planar graphs with maximum degree Delta\geq 9 are (\Delta+1)-edge-choosable -- short proof
We give a short proof of the following theorem due to Borodin~\cite{Bor90}. Every planar graph with maximum degree is -edge-choosable
An Improved Bound for the Linear Arboricity Conjecture
A forest is linear if all its components are paths. The linear arboricity
conjecture states that any graph of maximum degree can be
decomposed into at most linear forests. Here, we show
that admits a decomposition into at most linear forests provided is large enough. This improves a
recent result of Ferber, Fox and Jain. Moreover, our result also holds in a
more general list setting, where edges have (possibly different) sets of
permissible linear forests. The proof is based on a simple tweak of a
well-known technique in list edge-colouring, which was introduced by Kahn and
refined by Molloy and Reed
Pseudorandom hypergraph matchings
A celebrated theorem of Pippenger states that any almost regular hypergraph
with small codegrees has an almost perfect matching. We show that one can find
such an almost perfect matching which is `pseudorandom', meaning that, for
instance, the matching contains as many edges from a given set of edges as
predicted by a heuristic argument.Comment: 14 page
Edge-colouring graphs with local list sizes
The famous List Colouring Conjecture from the 1970s states that for every
graph the chromatic index of is equal to its list chromatic index. In
1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds
asymptotically. Our main result is a local generalization of Kahn's theorem.
More precisely, we show that, for a graph with sufficiently large maximum
degree and minimum degree , the following
holds: for every assignment of lists of colours to the edges of , such that
for
each edge , there is an -edge-colouring of . Furthermore, Kahn
showed that the List Colouring Conjecture holds asymptotically for linear,
-uniform hypergraphs, and recently Molloy generalized Kahn's original result
to correspondence colouring as well as its hypergraph generalization. We prove
local versions of all of these generalizations by showing a weighted version
that simultaneously implies all of our results.Comment: 22 page
A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree
A collection of graphs is \textit{nearly disjoint} if every pair of them
intersects in at most one vertex. We prove that if are nearly
disjoint graphs of maximum degree at most , then the following holds. For
every fixed , if each vertex is contained in
at most of the graphs , then the (list) chromatic number
of is at most . This result confirms a special
case of a conjecture of Vu and generalizes Kahn's bound on the list chromatic
index of linear uniform hypergraphs of bounded maximum degree. In fact, this
result holds for the correspondence (or DP) chromatic number and thus implies a
recent result of Molloy, and we derive this result from a more general list
coloring result in the setting of `color degrees' that also implies a result of
Reed and Sudakov.Comment: 14 pages with one-page appendix; this version adds Theorem 1.5 due to
L. Postl
Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond
The \emph{choice number} of a graph , denoted , is the minimum
integer such that for any assignment of lists of size to the vertices
of , there is a proper colouring of such that every vertex is mapped to
a colour in its list. For general graphs, the choice number is not bounded
above by a function of the chromatic number.
In this thesis, we prove a conjecture of Ohba which asserts that
whenever . We also prove a
strengthening of Ohba's Conjecture which is best possible for graphs on at most
vertices, and pose several conjectures related to our work.Comment: Master's Thesis, McGill Universit