The Backbone Coloring Problem for Bipartite Backbones

Abstract

Let $$G$$G be a simple graph, $$H$$H be its spanning subgraph and $$\lambda \ge 2$$??2 be an integer. By a $$\lambda $$?-backbone coloring of $$G$$G with backbone $$H$$H we mean any function $$c$$c that assigns positive integers to vertices of $$G$$G in such a way that $$|c(u)-c(v)|\ge 1$$|c(u)-c(v)|?1 for each edge $$uv\in E(G)$$uv?E(G) and $$|c(u)-c(v)|\ge \lambda $$|c(u)-c(v)|?? for each edge $$uv\in E(H)$$uv?E(H). The $$\lambda $$?-backbone chromatic number $$BBC_\lambda (G,H)$$BBC?(G,H) is the smallest integer $$k$$k such that there exists a $$\lambda $$?-backbone coloring $$c$$c of $$G$$G with backbone $$H$$H satisfying $$\max c(V(G))=k$$maxc(V(G))=k. A $$\lambda $$?-backbone coloring $$c$$c of $$G$$G with backbone $$H$$H is optimal if and only if $$\max c(V(G))=BBC_\lambda (G,H)$$maxc(V(G))=BBC?(G,H). In the paper we study the problem of finding optimal $$\lambda $$?-backbone colorings of complete graphs with bipartite backbones. We present a linear algorithm that is $$2$$2-approximate in general and $$1.5$$1.5-approximate if backbone is connected. Next we show a quadratic algorithm for backbones being trees that finds optimal $$\lambda $$?-backbone colorings provided $$\lambda $$? is large enough.