109 research outputs found
Hitting all maximum cliques with a stable set using lopsided independent transversals
Rabern recently proved that any graph with omega >= (3/4)(Delta+1) contains a
stable set meeting all maximum cliques. We strengthen this result, proving that
such a stable set exists for any graph with omega > (2/3)(Delta+1). This is
tight, i.e. the inequality in the statement must be strict. The proof relies on
finding an independent transversal in a graph partitioned into vertex sets of
unequal size.Comment: 7 pages. v4: Correction to statement of Lemma 8 and clarified proof
The covering radius problem for sets of perfect matchings
Consider the family of all perfect matchings of the complete graph
with vertices. Given any collection of perfect matchings of
size , there exists a maximum number such that if ,
then there exists a perfect matching that agrees with each perfect matching in
in at most edges. We use probabilistic arguments to give
several lower bounds for . We also apply the Lov\'asz local lemma to
find a function such that if each edge appears at most times
then there exists a perfect matching that agrees with each perfect matching in
in at most edges. This is an analogue of an extremal result
vis-\'a-vis the covering radius of sets of permutations, which was studied by
Cameron and Wanless (cf. \cite{cameron}), and Keevash and Ku (cf. \cite{ku}).
We also conclude with a conjecture of a more general problem in hypergraph
matchings.Comment: 10 page
A proof of the Erd\H{o}s-Faber-Lov\'asz conjecture
The Erd\H{o}s-Faber-Lov\'{a}sz conjecture (posed in 1972) states that the
chromatic index of any linear hypergraph on vertices is at most . In
this paper, we prove this conjecture for every large . We also provide
stability versions of this result, which confirm a prediction of Kahn.Comment: 39 pages, 2 figures; this version includes additional references and
makes two small corrections (definition of a useful pair in Section 5 and an
additional condition in the statement of Lemma 6.2
Recent developments in graph Ramsey theory
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress
- …