Abstract
Let \(G\) be a simple graph, \(H\) be its spanning subgraph and \(\lambda \ge 2\) be an integer. By a \(\lambda \)-backbone coloring of \(G\) with backbone \(H\) we mean any function \(c\) that assigns positive integers to vertices of \(G\) in such a way that \(|c(u)-c(v)|\ge 1\) for each edge \(uv\in E(G)\) and \(|c(u)-c(v)|\ge \lambda \) for each edge \(uv\in E(H)\). The \(\lambda \)-backbone chromatic number \(BBC_\lambda (G,H)\) is the smallest integer \(k\) such that there exists a \(\lambda \)-backbone coloring \(c\) of \(G\) with backbone \(H\) satisfying \(\max c(V(G))=k\). A \(\lambda \)-backbone coloring \(c\) of \(G\) with backbone \(H\) is optimal if and only if \(\max c(V(G))=BBC_\lambda (G,H)\). 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 \(2\)-approximate in general and \(1.5\)-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.
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Notes
If \(H\) is empty then \(BBC_\lambda (K_n,H)=n\) and any numbering of vertices of \(K_n\) with numbers \(1\), \(2\), ..., \(n\) is an optimal \(\lambda \)-coloring.
If \(r\) is optimal, we get a \(\lambda \)-backbone coloring \(c\) with \(\max c(V)=\lambda +1+\chi (K_n,H,q)\). Since \(H\) is connected, there are exactly two possible \(2\)-colorings of \(H\): \(q\) and \(1-q\). Our claim follows from Theorem 2 and the fact that \(\chi (K_n,H,q)=\chi (K_n,H,1-q)\) provided that \(H\) is bipartite.
It is easy to show that for \(\lambda \le n_0\) and \(H\) being a tree, \(BBC_\lambda (K_n,H)=n\) and our algorithm produces optimal \(\lambda \)-backbone coloring.
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This project has been partially supported by Narodowe Centrum Nauki under contract DEC-2011/02/A/ST6/00201.
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Janczewski, R., Turowski, K. The Backbone Coloring Problem for Bipartite Backbones. Graphs and Combinatorics 31, 1487–1496 (2015). https://doi.org/10.1007/s00373-014-1462-9
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DOI: https://doi.org/10.1007/s00373-014-1462-9