Backbone colorings for graphs: Tree and path backbones

Abstract

Abstract:We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1, 2, . . .} of G in which the colors assigned to adjacent vertices in H differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number χ(G); for path backbones this factor is roughly 3 2. We show that the computational complexity of the problem "Given a graph G, a spanning tree T of G, and an integer , is there a backbone coloring for G and T with at most colors?" jumps from polynomial to NP-complete between = 4 (easy for all spanning trees) and = 5 (difficult even for spanning paths). We finish the paper by discussing some open problems.

Steinberg-like theorems for backbone colouring

“…It is a q-backbone colouring for (G, H) if f is a proper colouring of G and |f (u)−f (v)| ≥ q for all edges uv ∈ E(H). The chromatic number χ(G) is the smallest integer k for which there exists a proper k-colouring of G. The q-backbone chromatic number of (G, H), denoted by BBC q (G, H), is the smallest integer k for which there exists a q-backbone k-colouring of (G, H) [3].…”

“…Inequality (1) and the Four-Colour Theorem [1] imply that for any planar graph G and spanning subgraph H, BBC q (G, H) ≤ 3q + 1. However, for q = 2, Broersma et al [3] conjectured that this is not best possible if the backbone is a forest. Conjecture 1.…”

Backbone colorings along stars and matchings in split graphs: their span is close to the chromatic number

“…In [7] backbone colorings are introduced, motivated and put into a general framework of coloring problems related to frequency assignment. Graphs are used to model the topology and interference between transmitters (receivers, base stations): the vertices represent the transmitters; two vertices are adjacent if the corresponding transmitters are so close (or so strong) that they are likely to interfere if they broadcast on the same or 'similar' frequency channels.…”

“…This has led to various different types of coloring problems in graphs, depending on different ways to model the level of interference, the notion of similar frequency channels, and the definition of acceptable level of interference (See, e.g., [17], [22]). We refer to [6] and [7] for an overview of related research, but we repeat the general framework and some of the related research here for convenience and background.…”

Performance Optimization of Network Protocols for IEEE 802.11s-based Smart Grid Communications

“…Therefore, the proposed approach used an MST of the network topology rooted at the gateway g and assigns a time slot t j to k SMs based on their locations in the ST. In the context of VCPs, instead of assigning colors to graph G(V ,E), the proposed approach strives to assign colors to vertices of a spanning tree subgraph T (V ,E T ) of G. This vertex coloring of a spanning tree is known as backbone coloring [97].…”