Backbone colorings for graphs: Tree and path backbones

  • Authors:

  • Hajo Broersma;Fedor V. Fomin;Petr A. Golovach;Gerhard J. Woeginger

  • Affiliations:

  • Department of Computer Science, Durham University, Durham, UK;Department of Informatics, University of Bergen, N-5020 Bergen, Norway;Faculty of Mathematics, Syktyvkar State University, 167001 Syktyvkar, Russia;Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, the Netherlands

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2007

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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 ${{3}\over{2}}$. We show that the computational complexity of the problem “Given a graph G, a spanning tree T of G, and an integer ℓ, is there a backbone coloring for G and T with at most ℓ colors?” jumps from polynomial to NP-complete between ℓ = 4 (easy for all spanning trees) and ℓ = 5 (difficult even for spanning paths). We finish the paper by discussing some open problems. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 137–152, 2007 Part of the work was done while FVF and PAG were visiting the University of Twente.