On the total chromatic edge stability number and the total chromatic subdivision number of graphs

A proper total coloring of a graph G is an assignment of colors to the vertices and edges of G (together called the elements of G) such that neighbored elements-two adjacent vertices or two adjacent edges or a vertex and an incident edge-are colored differently. The total chromatic number chi ''(G) of G is defined as the minimum number of colors in a proper total coloring of G. In this paper, we study the stability of the total chromatic number of a graph with respect to two operations, namely removing edges and subdividing edges, which leads to the following two invariants. (i) The total chromatic edge stability number or chi ''-edge stability number es(chi '')(G) is the minimum number of edges of G whose removal results in a graph H subset of G with chi ''(H) not equal chi ''(G) or with E(H) = empty set. (ii) The total chromatic subdivision number or chi ''-subdivision number sd(chi '')(G) is the minimum number of edges of G whose subdivision results in a graph H subset of G with chi ''(H) not equal chi ''(G) or with E(H) not equal empty set. We prove general lower and upper bounds for es(chi '')(G). Moreover, we determine sd(chi '')(G) and sd(chi '')(G) for some classes of graphs.