Conditional and Unique Coloring of Graphs (revised resubmission)
Abstract
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 \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 r(G). For different values of r we first give results (exact values or bounds for r(G) depending on r) related to the conditional coloring of graphs. Then we obtain 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 unique conditional colorability and give some related results.