On the Total Chromatic Edge Stability Number and the Total Chromatic Subdivision Number of Graphs
Arnfried Kemnitz, Massimiliano Marangio · Discrete Mathematics Letters · 2022
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 esor with E(H) = ∅.We prove general lower and upper bounds for es χ (G).Moreover, we determine es χ (G) and sd χ (G) for some classes of graphs.