4 research outputs found
An asymptotic bound for the strong chromatic number
The strong chromatic number of a graph on
vertices is the least number with the following property: after adding isolated vertices to and taking the union with any
collection of spanning disjoint copies of in the same vertex set, the
resulting graph has a proper vertex-colouring with colours.
We show that for every and every graph on vertices with
, , which is
asymptotically best possible.Comment: Minor correction, accepted for publication in Combin. Probab. Compu
On sufficient conditions for Hamiltonicity in dense graphs
We study structural conditions in dense graphs that guarantee the existence
of vertex-spanning substructures such as Hamilton cycles. It is easy to see
that every Hamiltonian graph is connected, has a perfect fractional matching
and, excluding the bipartite case, contains an odd cycle. Our main result in
turn states that any large enough graph that robustly satisfies these
properties must already be Hamiltonian. Moreover, the same holds for embedding
powers of cycles and graphs of sublinear bandwidth subject to natural
generalisations of connectivity, matchings and odd cycles.
This solves the embedding problem that underlies multiple lines of research
on sufficient conditions for Hamiltonicity in dense graphs. As applications, we
recover and establish Bandwidth Theorems in a variety of settings including
Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type
conditions, locally dense and inseparable graphs, multipartite graphs as well
as robust expanders
Extremal results in hypergraph theory via the absorption method
The so-called "absorbing method" was first introduced in a systematic way by Rödl, Ruciński and Szemerédi in 2006, and has found many uses ever since. Speaking in a general sense, it is useful for finding spanning substructures of combinatorial structures. We establish various results of different natures, in both graph and hypergraph theory, most of them using the absorbing method:
1. We prove an asymptotically best-possible bound on the strong chromatic number with respect to the maximum degree of the graph. This establishes a weak version of a conjecture of Aharoni, Berger and Ziv.
2. We determine asymptotic minimum codegree thresholds which ensure the existence of tilings with tight cycles (of a given size) in uniform hypergraphs. Moreover, we prove results on coverings with tight cycles.
3. We show that every 2-coloured complete graph on the integers contains a monochromatic infinite path whose vertex set is sufficiently "dense" in the natural numbers. This improves results of Galvin and Erdős and of DeBiasio and McKenney