Total coloring of quasi-line graphs and inflated graphs

A total coloring of a graph is an assignment of colors to all the elements (vertices and edges) of the graph such that no two adjacent or incident elements receive the same color. A claw-free graph is a graph that does not have [Formula: see text] as an induced subgraph. Quasi-line and inflated graphs are two well-known classes of claw-free graphs. In this paper, we prove that the quasi-line and inflated graphs are totally colorable. In particular, we prove the tight bound of the total chromatic number of some classes of quasi-line graphs and inflated graphs.

Total colorings of circulant graphs

The total chromatic number [Formula: see text] is the least number of colors needed to color the vertices and edges of a graph [Formula: see text] such that no incident or adjacent elements (vertices or edges) receive the same color. Behzad and Vizing proposed a well-known total coloring conjecture (TCC): [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. For the powers of cycles, Campos and de Mello proposed the following conjecture: Let [Formula: see text] denote the graphs of powers of cycles of order [Formula: see text] and length [Formula: see text] with [Formula: see text]. Then, [Formula: see text] In this paper, we prove the Campos and de Mello’s conjecture for some classes of powers of cycles. Also, we prove the TCC for complement of powers of cycles.

Local Super Antimagic Total Labeling for Vertex Coloring of Graphs

Let G=(V,E) be a graph with vertex set V and edge set E. A local antimagic total vertex coloring f of a graph G with vertex-set V and edge-set E is an injective map from V∪E to {1,2,…,|V|+|E|} such that if for each uv∈E(G) then w(u)≠w(v), where w(u)=∑uv∈E(G)f(uv)+f(u). If the range set f satisfies f(V)={1,2,…,|V|}, then the labeling is said to be local super antimagic total labeling. This labeling generates a proper vertex coloring of the graph G with the color w(v) assigning the vertex v. The local super antimagic total chromatic number of graph G, χlsat(G) is defined as the least number of colors that are used for all colorings generated by the local super antimagic total labeling of G. In this paper we investigate the existence of the local super antimagic total chromatic number for some particular classes of graphs such as a tree, path, cycle, helm, wheel, gear, sun, and regular graphs as well as an amalgamation of stars and an amalgamation of wheels.

Splitting on classes of helm graphs and its equitabletotal chromatic number

The equitable total coloring of a graph $G$ is the different colors used to color all the vertices and edges of $G$, in the order that adjacent vertices and edges are assigned with least different $k$-colors and can be partitioned into colors sets which differ by maximum one. The minimum of $k$-colors required is known as the equitable total chromatic number. In this paper the splitting graph of Helm and Closed Helm graph is constructed and its equitable total chromatic number is acquired.