On the Total Chromatic Edge Stability Number and the Total Chromatic Subdivision Number of Graphs

Abstract

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 χ′′(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 χ′′-edge stability number esχ′′(G) is the minimum number of edges of G whose removal results in a graphH ⊆ G with χ′′(H) 6= χ′′(G) or with E(H) = ∅. (ii) The total chromatic subdivision number or χ′′-subdivision number sdχ′′(G) is the minimum number of edges of G whose subdivision results in a graph H ⊆ G with χ′′(H) 6= χ′′(G) or with E(H) = ∅. We prove general lower and upper bounds for esχ′′(G). Moreover, we determine esχ′′(G) and sdχ′′(G) for some classes of graphs.