Relations of vertex distinguishing total chromatic numbers between a subgraph and its supergraph

Graph theory is the historical foundation of the science of networks and the basis of information science. Various colorings of graphs are very important problem in the research of graph theory. Let f be a proper k-total coloring of a graph G  . For any vertex u∈V(G)u∈V(G), let Cf(u)Cf(u) denote the set of colors of vertex u   and its incident edges. If Cf(u)≠Cf(v) for any two distinct vertices u and v   of V(G)V(G), then f is called a k-vertex distinguishing total coloring of G. The minimum integer k for which there exists a k-vertex distinguishing total coloring of G is called the vertex distinguishing total chromatic number of G. Relations of vertex distinguishing total chromatic numbers between a subgraph and its supergraph are discussed in this paper. We will give the result: for each positive integer r, there exists a supergraph G with maximum degree r and its subgraph H, such that the vertex distinguishing total chromatic number of H is greater than the vertex distinguishing total chromatic number of G; We will also discuss the relation of the vertex distinguishing total chromatic number between a supergraph G   and its subgraph G–eG–e, where e is an edge of G; We will obtain the sufficient conditions for the vertex distinguishing total chromatic number of a subgraph of a complete graph K with odd order being equal to the vertex distinguishing total chromatic number of supergraph K. For common (proper) vertex chromatic number and (proper) edge chromatic number, this number of subgraph is not greater than that of supergraph. So the research results in this paper are very interesting and of great significance.