Entire chromatic number and \(\Delta\)-matching of outerplane graphs (Q815128)

A plane graph \(G\) is \(k\)-entire colorable if the elements of \(V(G)\cup E(G)\cup F(G)\) can be colored with \(k\) colors such that any two adjacent or incident elements receive different colors. The entire chromatic number \(\chi_{\text{vef}}(G)\) of \(G\) is the minimum number \(k\) such that \(G\) is \(k\)-entire colorable. Let \(G\) be an outerplane graph with maximum degree \(\Delta\) and the entire chromatic number \(\chi_{\text{vef}}(G)\). This paper proves that if \(\Delta\geq6\), then \(\Delta+1\leq\chi_{\text{vef}}(G)\leq\Delta+2\), and \(\chi_{\text{vef}}(G)\leq\Delta+1\) if and only if \(G\) has a matching consisting of some inner edges which covers all its vertices of maximum degree.