Entire Chromatic Number of \(1\)-Tree

Abstract

The entire chromatic number \(\chi_c(G)\) of a plane graph \(G(V, E, F)\) is the minimum number of colors such that any two distinct adjacent or incident elements receive different colors in \(V(G) \cup E(G) \cup F(G)\). A plane graph \(G\) is called a \(1\)-tree if there exists a vertex \(u \in V(G)\) such that \(G – u\) is a forest. In this paper, it is proved that if \(G\) is a \(2\)-connected \(1\)-tree with \(\Delta(G) \geq 6\), then the entire chromatic number of \(G\) is \(\Delta(G) + 1\), where \(\Delta(G)\) is the maximum degree of \(G\).