Abstract
Graph Theory A natural generalization of graph colouring involves taking colours from a metric space and insisting that the endpoints of an edge receive colours separated by a minimum distance dictated by properties of the edge. In the q-backbone colouring problem, these minimum distances are either q or 1, depending on whether or not the edge is in the backbone. In this paper we consider the list version of this problem, with particular focus on colours in ℤp - this problem is closely related to the problem of circular choosability. We first prove that the list circular q-backbone chromatic number of a graph is bounded by a function of the list chromatic number. We then consider the more general problem in which each edge is assigned an individual distance between its endpoints, and provide bounds using the Combinatorial Nullstellensatz. Through this result and through structural approaches, we achieve good bounds when both the graph and the backbone belong to restricted families of graphs.
Highlights
All graphs considered in this paper are simple
It is a proper colouring if |f (u) − f (v)| ≥ 1
We explore similar restrictions shortly, but first we define a list colouring analogue
Summary
All graphs considered in this paper are simple. Let G = (V, E) be a graph, and let H = (V, E(H)) be a subgraph of G, called the backbone. The q-backbone chromatic number BBCq(G, H) is the smallest integer k for which there exists a q-backbone k-colouring of (G, H). The circular q-backbone chromatic number of a graph pair (G, H), denoted CBCq(G, H), is the minimum k such that (G, H) admits a circular q-backbone k-colouring. CBCq(G, H) ≤ q · χ(G) for every subgraph H of G This bound is tight if H contains a complete graph of size χ(G). We explore similar restrictions shortly, but first we define a list colouring analogue Such an analogue is motivated by the fact that backbone colouring models a special case of radio channel assignment: in such problem, because of technical reasons or dynamicity, the set of channels available (i.e. colours) very often varies from transmitter (i.e. vertex) to transmitter
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