Conditional and Unique Coloring of Graphs (revised resubmission)

For integers k > 0 and 0 < r ≤ ∆ (where r ≤ 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 rth 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 pa-rameterized 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 unique conditional colorability and give some related results.