54 research outputs found
Optimal path and cycle decompositions of dense quasirandom graphs
Motivated by longstanding conjectures regarding decompositions of graphs into
paths and cycles, we prove the following optimal decomposition results for
random graphs. Let be constant and let . Let be
the number of odd degree vertices in . Then a.a.s. the following hold:
(i) can be decomposed into cycles and a
matching of size .
(ii) can be decomposed into
paths.
(iii) can be decomposed into linear forests.
Each of these bounds is best possible. We actually derive (i)--(iii) from
`quasirandom' versions of our results. In that context, we also determine the
edge chromatic number of a given dense quasirandom graph of even order. For all
these results, our main tool is a result on Hamilton decompositions of robust
expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte
On the classification problem for split graphs
Abstract
The Classification Problem is the problem of deciding whether a simple graph has chromatic index equal to Δ or Δ+1. In the first case, the graphs are called Class 1, otherwise, they are Class 2. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs are a subclass of chordal graphs. Figueiredo at al. (J. Combin. Math. Combin. Comput. 32:79–91, 2000) state that a chordal graph is Class 2 if and only if it is neighborhood-overfull. In this paper, we give a characterization of neighborhood-overfull split graphs and we show that the above conjecture is true for some split graphs
Towards the Overfull Conjecture
Let be a simple graph with maximum degree denoted as . An
overfull subgraph of is a subgraph satisfying the condition . In 1986, Chetwynd and Hilton
proposed the Overfull Conjecture, stating that a graph with maximum degree
has chromatic index equal to if and
only if it does not contain any overfull subgraph. The Overfull Conjecture has
many implications. For example, it implies a polynomial-time algorithm for
determining the chromatic index of graphs with , and implies several longstanding conjectures in the area of
graph edge colorings. In this paper, we make the first improvement towards the
conjecture when not imposing a minimum degree condition on the graph: for any
, there exists a positive integer such
that if is a graph on vertices with , then the Overfull Conjecture holds for . The previous
best result in this direction, due to Chetwynd and Hilton from 1989, asserts
the conjecture for graphs with .Comment: arXiv admin note: text overlap with arXiv:2205.08564,
arXiv:2105.0528
Graph edge coloring and a new approach to the overfull conjecture
The graph edge coloring problem is to color the edges of a graph such that adjacent edges receives different colors. Let be a simple graph with maximum degree . The minimum number of colors needed for such a coloring of is called the chromatic index of , written . We say is of class one if , otherwise it is of class 2. A majority of edge coloring papers is devoted to the Classification Problem for simple graphs. A graph is said to be \emph{overfull} if . Hilton in 1985 conjectured that every graph of class two with contains an overfull subgraph with . In this thesis, I will introduce some of my researches toward the Classification Problem of simple graphs, and a new approach to the overfull conjecture together with some new techniques and ideas
The chromatic index of graphs of high maximum degree
AbstractIn this paper, we give sufficient conditions for simple graphs to be class 1. These conditions mainly depend on the edge-connectivity, maximum degree and the number of vertices of maximum degree of a graph. Using these conditions, we can extend various results of Chetwynd and Hilton, and Niessen and Volkmann
Some applications of matching theorems
PhDThis thesis contains the results of two investigations. The rst concerns the 1-
factorizability of regular graphs of high degree. Chetwynd and Hilton proved in
1989 that all regular graphs of order 2n and degree 2n where
> 1
2 (
p
7 1) 0:82288
are 1-factorizable. We show that all regular graphs of order 2n and degree 2n
where is greater than the second largest root of
4x6 28x5 71x4 + 54x3 + 88x2 62x + 3
( 0:81112) are 1-factorizable. It is hoped that in the future our techniques will
yield further improvements to this bound. In addition our study of barriers in
graphs of high minimum degree may have independent applications.
The second investigation concerns partial latin squares that satisfy Hall's Condition.
The problem of completing a partial latin square can be viewed as a listcolouring
problem in a natural way. Hall's Condition is a necessary condition for
such a problem to have a solution. We show that for certain classes of partial latin
square, Hall's Condition is both necessary and su cient, generalizing theorems of
Hilton and Johnson, and Bobga and Johnson. It is well-known that the problem
of deciding whether a partial latin square is completable is NP-complete. We
show that the problem of deciding whether a partial latin square that is promised
to satisfy Hall's Condition is completable is NP-hard
- …