On the \(d\)-distance face chromatic number of plane graphs
Let \(G\) be a finite connected plane graph, and let \(F(G)\) be the set of all faces of \(G\). The distance of two faces \(f_1\), \(f_2\) of \(G\) is the minimum value of \(d(x_1,x_2)\) for \(x_i\in f_i\). For an integer \(d\leq 0\), the \(d\)-distance face \(k\)-colouring of \(G\) is a map \(\varphi: F(G)\to\{1,\dots, k\}\) such that if \(f_1\neq f_2\) and \(d(f_1,f_2)\leq d\) then \(\varphi(f_1)\neq\varphi(f_2)\). The minimum \(k\) for which such a map exists is called the \(d\)-distance face chromatic number, and is denoted by \(df_c(G)\). In this paper, an upper bound for \(df_c(G)\) is given when the maximum degree is greater than or equal to 8. Some known results for \(d=0\) are obtained as corollaries.
scientific article; zbMATH DE number 1019060
scientific article