Rainbow Neighbourhood Equate Number of Graphs

In this paper, a new invariant of a graph namely, the rainbow neighbourhood equate number of a graph $G$ denoted by $ren(G)$ is introduced. It is defined to be the minimum number of vertices whose removal results in a subgraph that admits a $J$-colouring. The new notions of chromatic degree of a vertex $d_{\chi }(v)$, the maximum and minimum chromatic degrees of $G$ denoted, ${\mathrm{\Delta }}_{\chi }(G)$ and ${\delta }_{\chi }(G)$ respectively, are also introduced. The chromatic diameter of $G$ denoted, $d(G,\chi )$ is introduced as well. The study of $ren(G)$ appears to be very complex for graphs in general so for now, only introductory results will be presented. Finally, the concept of a chromatic degree sequence is proposed as a new research direction.