Total Chromatic Number of Folded Hypercubes

Abstract

A total coloring of a simple graph \(G\) is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color.The minimum number of colors required for a proper total coloring of \(G\) is called the total chromatic number of \(G\) and denoted by \(\chi_t(G)\). The Total Coloring Conjecture (TCC) states that for every simple graph \(G\),\(\Delta(G) + 1 \leq \chi_t(G) \leq \Delta(G) + 2.\) \(G\) is called Type \(1\) (resp. Type \(2\)) if \(\chi_t(G) = \Delta(G) +1\) (resp. \(\chi_t(G) = \Delta(G) + 2\)). In this paper, we prove that the folded hypercubes \(FQ_n\), is of Type \(1\) when \(n \geq 4\).