73 research outputs found
Proof of a conjecture on induced subgraphs of Ramsey graphs
An n-vertex graph is called C-Ramsey if it has no clique or independent set
of size C log n. All known constructions of Ramsey graphs involve randomness in
an essential way, and there is an ongoing line of research towards showing that
in fact all Ramsey graphs must obey certain "richness" properties
characteristic of random graphs. More than 25 years ago, Erd\H{o}s, Faudree and
S\'{o}s conjectured that in any C-Ramsey graph there are
induced subgraphs, no pair of which have the same
numbers of vertices and edges. Improving on earlier results of Alon, Balogh,
Kostochka and Samotij, in this paper we prove this conjecture
An Approximate Version of the Tree Packing Conjecture via Random Embeddings
We prove that for any pair of constants a>0 and D and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most D, and with at most n(n-1)/2 edges in total packs into the complete graph of order (1+a)n. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof
Vertex covering with monochromatic pieces of few colours
In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every -colouring of the
edges of , there is a vertex cover by monochromatic paths of
the same colour, which is optimal up to a constant factor. The main goal of
this paper is to study the natural multi-colour generalization of this problem:
given two positive integers , what is the smallest number
such that in every colouring of the edges of with
colours, there exists a vertex cover of by
monochromatic paths using altogether at most different colours? For fixed
integers and as , we prove that , where is the chromatic number of
the Kneser gr aph . More generally, if one replaces by
an arbitrary -vertex graph with fixed independence number , then we
have , where this time around is the
chromatic number of the Kneser hypergraph . This
result is tight in the sense that there exist graphs with independence number
for which . This is in sharp
contrast to the case , where it follows from a result of S\'ark\"ozy
(2012) that depends only on and , but not on
the number of vertices. We obtain similar results for the situation where
instead of using paths, one wants to cover a graph with bounded independence
number by monochromatic cycles, or a complete graph by monochromatic
-regular graphs
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