Stability in Respect of Chromatic Completion of Graphs

In an improper colouring an edge $uv$ for which, $c(u)=c(v)$ is called a \emph{bad edge}. The notion of the \emph{chromatic completion number} of a graph $G$ denoted by $\zeta(G),$ is the maximum number of edges over all chromatic colourings that can be added to $G$ without adding a bad edge. We introduce stability of a graph in respect of chromatic completion. We prove that the set of chromatic completion edges denoted by $E_\chi(G),$ which corresponds to $\zeta(G)$ is unique if and only if $G$ is stable in respect of chromatic completion. Thereafter, chromatic completion and stability is discussed in respect of Johan colouring. The difficulty of studying chromatic completion with regards to graph operations is shown by presenting results for two elementary graph operations.