ﻻ يوجد ملخص باللغة العربية
Given $varepsilon>0$, there exists $f_0$ such that, if $f_0 le f le Delta^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $Delta$ in which the neighbourhood of every vertex in $G$ spans at most $Delta^2/f$ edges, (i) an independent set of $G$ drawn uniformly at random has at least $(1/2-varepsilon)(n/Delta)log f$ vertices in expectation, and (ii) the fractional chromatic number of $G$ is at most $(2+varepsilon)Delta/log f$. These bounds cannot in general be improved by more than a factor $2$ asymptotically. One may view these as strong
Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs.
This paper provides a survey of methods, results, and open problems on graph and hypergraph colourings, with a particular emphasis on semi-random `nibble methods. We also give a detailed sketch of some aspects of the recent proof of the ErdH{o}s-Faber-Lov{a}sz conjecture.
Switches are operations which make local changes to the edges of a graph, usually with the aim of preserving the vertex degrees. We study a restricted set of switches, called triangle switches. Each triangle switch creates or deletes at least one tri
An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation emph{proper} if neighbouring vertices have different in-degrees. The proper orientation nu
The Ising antiferromagnet is an important statistical physics model with close connections to the {sc Max Cut} problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry break