Backbone Colorings for Networks

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 . For the special case that G is a split graph the difference from χ(G) is at most an additive constant 2 or 1, for tree backbones and path backbones, respectively. 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).

Backbone colorings for graphs: Tree and path backbones

“…Split graphs were introduced by Hammer & Földes [16]; see also the book [13] by Golumbic. Split graphs are perfect graphs, and hence satisfy χ(G) = ω(G), where ω(G) is the size of a largest clique in G. It is known (see [5] for detailed information), that for every spanning tree T in a split graph G, bbc(G, T ) ≤ χ(G) + 2. Also if ω(G) = 3, then for every Hamiltonian path P in G, bbc(G, P) ≤ χ(G) + 1, and if ω(G) = 3, then bbc(G, P) ≤ 5.…”

“…The work presented here is a full version of an extended abstract that appeared in the Proceedings of WG 2003 [5]. It is motivated by the general framework for coloring problems related to frequency assignment.…”

On the Existence of Tree Backbones that Realize the Chromatic Number on a Backbone Coloring

“…This gives an upper bound of 2k − 1 for T k . In [2], they proved that this is actually best possible. Theorem 1.Broersma et al [2].…”

“…This parameter was first introduced by Broersma et al. as a model for the frequency assignment problem where certain channels of communication are more demanding than others. In their seminal work, they only considered and they were interested in finding out how far away from can be in the worst case.…”

Backbone colourings of graphs

“…Backbone colouring was firstly motivated by the colouring problems related to the frequency assignment problem and was introduced by Broersma et al [2]. Their main result was that for any connected graph G and any spanning tree T of G,…”