Conditional and Unique Coloring of Graphs (revised resubmission)

TL;DR

This paper studies a new type of graph coloring called conditional $(k,r)$-coloring, providing exact values and bounds for various graphs, and introduces the concept of unique conditional colorability.

Contribution

It introduces the concept of conditional $(k,r)$-coloring, derives results for specific graphs, and defines the notion of unique conditional colorability.

Findings

Determined $ ext{chi}_r(G)$ for several parameterized graphs.

Established bounds and exact values for conditional chromatic numbers.

Introduced and analyzed the concept of unique conditional colorability.

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.