Consecutive List Colouring and a New Graph Invariant

Abstract

We consider a variation of the list colouring problem in which the lists are required to be sets of consecutive integers, and the colours assigned to adjacent vertices must differ by at least a fixed integer s. We introduce and investigate a new parameter τ (G) of a graph G, called the consecutive choosability ratio and defined to be the ratio of the required list size to the separation s in the limit as s → ∞.We show that the above limit exists and that, for finite graphs G, τ (G) is rational and is a refinement of the chromatic number χ(G). We provide general bounds on τ (G), and determine its value for various classes of graphs including bipartite graphs, circuits, wheels and balanced complete multipartite graphs. Finally, we explore relationships between τ (G) and the circular chromatic number χc(G).

Circular choosability is rational

“…In his thesis [13] and in [14] the second author introduced and studied the consecutive choosability ratio τ (G) which is defined similarly to σ (G) with the difference that all lists are intervals. He showed that τ (G) can be written as τ (G) = a/b with b n. A very similar concept is the circular consecutive choosability, introduced by Lin et al [3] and studied further by Norine et al [6] and Pan and Zhu [9].…”

Circular consecutive choosability of k-choosable graphs

“…Besides the choice number, several variations of choosability have also been studied in the literature. One of them is the consecutive choosability, introduced by Waters [10], in which the list-assignment for each vertex is a set of consecutive integers. Another variation, called circular choosability, is motivated by the circular coloring of graphs.…”

“…Parallel to the investigation of the consecutive variation of choosability [10], it is natural to consider the consecutive variation of circular choosability in which the list of each vertex is a single closed interval on S(r). This is a notion first introduced and studied by Lin et al [5].…”