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- W1843060384 abstract "For integers $k>0$ and $0<r leq Delta$ (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 $chi_r(G)$. For different values of $r$ we first give results (exact values or bounds for $chi_r(G)$ depending on $r$) related to the conditional coloring of graphs. Then we obtain $chi_r(G)$ 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." @default.
- W1843060384 created "2016-06-24" @default.
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- W1843060384 date "2012-01-30" @default.
- W1843060384 modified "2023-09-27" @default.
- W1843060384 title "Conditional and Unique Coloring of Graphs (revised resubmission)" @default.
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