The total chromatic number of Pseudo-Halin graphs with lower degree

The total chromatic number χT(G)χT(G) of a graph GG is the least number of colors needed to color the vertices and the edges of GG such that no adjacent or incident elements receive the same color. The Total Coloring Conjecture(TCC) states that for every simple graph GG, χT(G)≤Δ(G)+2χT(G)≤Δ(G)+2. In this paper, we show that χT(G)=Δ(G)+1χT(G)=Δ(G)+1 for all pseudo-Halin graphs with Δ(G)=4Δ(G)=4 and 5.