Backbone coloring of planar graphs without special circles
Given a graph \(G = (V,E)\) and a spanning subgraph \(H\) of \(G\) (the backbone of \(G\)), a backbone \(k\)-coloring of \((G,H)\) is a proper vertex coloring \(f: V \rightarrow \{1, 2, \ldots, k\}\) of \(G\) such that \(|f(u) - f(v)| \geq 2\) if \(uv \in E(H)\) and \(|f(u) - f(v)| \geq 1\) if \(uv \in E(G)\setminus E(H)\). The authors prove that any connected planar graph \(G\) without adjacent triangles and without chordless \(6\)-cycles or chordless \(7\)-cycles admits a spanning tree \(T\) such that \((G,T)\) has a backbone \(4\)-coloring. A related result [\textit{W. Wang, Y. Bu, M. Montassier} and \textit{A. Raspaud}, ``On backbone coloring of graphs,'' J. Comb. Optim. 23, 79--93 (2012)] is as follows: Any connected non-bipartite planar graph \(G\) without cycles of length less than \(10\) admits a spanning tree \(T\) such that \((G,T)\) has a backbone \(4\)-coloring.
scientific article; zbMATH DE number 5986980