Conditional and Unique Coloring of Graphs (revised resubmission)

P. V. Subba Reddy, K. V. Iyer

For integers $k>0$ and $0<r\le \Delta$ (where $r\le k$), a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of…