A Note on Vertex List Colouring

Abstract

Let k be a positive integer and let G be a graph. Suppose a list S(v) of positive integers is assigned to each vertex v, such that(1) [mid ]S(v)[mid ] = 2k for each vertex v of G, and(2) for each vertex v, and each c ∈ S(v), the number of neighbours w of v for which c ∈ S(w) is at most k.Then we prove that there exists a proper vertex colouring f of G such that f(v) ∈ S(v) for each v ∈ V(G). This proves a weak version of a conjecture of Reed.

On finite simple groups and Kneser graphs

“…By the following result (see also [12] and [22]), it is sufficient to show that 2 times the degree of any vertex in is at most (n − r − 1)!. If {v, w} is an edge in , then v and w generate a transitive subgroup of Sym( ) not containing Alt( ).…”

The Moser-Tardos Framework with Partial Resampling

“…Given a partition of the vertices of an undirected graph G = (V, E) into blocks (or classes), a transversal is a subset of the vertices, one chosen from each block. An independent transversal, or independent system of representatives, is a transversal that is also an independent set in G. The study of independent transversals was initiated by Bollobás, Erdős & Szemerédi [6], and has received a considerable amount of attention (see, e.g., [1,2,3,18,19,21,26,34,35]). Furthermore, such transversals serve as building blocks for other graph-theoretic parameters such as the linear arboricity and the strong chromatic number [2,3].…”

“…Haxell shows that a sufficient condition is b ≥ 2∆ [18]; this condition is existentially optimal, in the sense that b ≥ 2∆ − 1 is not always admissible [21,35,34]. The work of [19] gives a similar criterion of b ≥ ∆ + ∆/r for the existence of a transversal which induces no connected component of size > r. (Here r = 1 corresponds to independent transversals.)…”

“…The proof of [18] adds vertices one-by-one to partial transversals, which depends very heavily on bounding the maximum degree of any vertex. It is also highly non-constructive.…”

Constraint satisfaction, packet routing, and the lovasz local lemma

“…Nevertheless, this yields a pathway to proving the theorem. We apply the assignment LLL to improve bounds of [1,3,8,15,16,19,20,25,31]. However, our approach here does not appear to lead to an algorithmic counterpart of our assignment LLL.…”