Total Coloring in Some Split-Comparability Graphs

Abstract

A total coloring for a graph G is an assignment of colors to the edges and vertices of G such that any pair of adjacent or incident elements have different colors. The least number of colors for which G has a total coloring is denoted χ ′′ (G). It is known that splitcomparability graphs have χ ′′ (G) at most ∆(G) + 2. In this work we show that certain split-comparability graphs with odd maximum degree have χ ′′ (G) = ∆(G) + 1.