The computational complexity of the backbone coloring problem for bounded-degree graphs with connected backbones - Janczewski, Robert; Turowski, Krzysztof (Information processing letters, 2015)
Given a graph G, a spanning subgraph H of G and an integer λ≥2, a λ-backbone coloring of G with backbone H is a proper vertex coloring of G using colors 1,2,…, in which the color difference between vertices adjacent in H is greater than or equal to λ. The backbone coloring problem is that of finding such a coloring whose maximum color does not exceed a given limit k. In this paper, we study the backbone coloring problem for bounded-degree graphs with connected backbones and we give a complete computational complexity classification of this problem. We present a polynomial algorithm for optimal backbone coloring for subcubic graphs with arbitrary backbones. We also prove that the backbone coloring problem for graphs with arbitrary backbones and with fixed maximum degree (at least 4) is NP-complete. Furthermore, we show that for the special case of graphs with fixed maximum degree at least 5 and λ≥4 the problem remains NP-complete even for spanning tree backbones. •The backbone coloring problem is solvable in quadratic time for subcubic graphs.•The backbone coloring problem is NP-hard for graphs with degree greater than 4.•The backbone coloring problem is NP-hard for tree backbones with bounded degree.