Total Colouring of New Classes of Subcubic graphs

Abstract

The total chromatic number of a graph G , denoted χ ′′ ( G ), is the least number of colours needed to colour the vertices and the edges of G such that no incident or adjacent elements (vertices or edges) receive the same colour. The popular Total Colouring Conjecture (TCC) posed by Behzad states that, for every simple graph G , χ ′′ ( G ) ≤ ∆( G ) + 2. In this paper, we prove that the total chromatic number for a family of subcubic graphs called cube connected paths and also for a class of subcubic graphs having the property that the vertices are covered by independent triangles are exactly ∆( G ) + 1. More precisely, these two families of subcubic graphs are shown to be Type 1 graphs.