Conditional and Unique Coloring of Graphs (revised resubmission)

Abstract

For integers $k>0$ and $0<r \leq Δ$ (where $r \leq k$), 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 degree $d(v)$ in $G$ is adjacent to vertices with at least $\min\{r, d(v)\}$ differently colored neighbors. The smallest integer $k$ for which a graph $G$ has a conditional $(k,r)$-coloring is called the $r$th order conditional chromatic number, denoted by $χ_r(G)$. For different values of $r$ we first give results (exact values or bounds for $χ_r(G)$ depending on $r$) related to the conditional coloring of graphs. Then we obtain $χ_r(G)$ of certain parameterized graphs viz., windmill graph, line graph of windmill graph, middle graph of friendship graph, middle graph of a cycle, line graph of friendship graph, middle graph of complete $k$-partite graph, middle graph of a bipartite graph and gear graph. Finally we introduce \emph{unique conditional colorability} and give some related results.