Conditional and Unique Coloring of Graphs

TL;DR

This paper studies a specialized graph coloring method called conditional coloring, introduces the concept of unique conditional colorability, and provides bounds and exact values for the associated chromatic number for various graphs.

Contribution

It presents new bounds and exact values for the conditional chromatic number and introduces the concept of unique conditional colorability in graph theory.

Findings

Derived bounds for $ ext{chi}_r(G)$ for specific graphs.

Calculated exact values of $ ext{chi}_r(G)$ in certain cases.

Introduced and explored the concept of unique conditional colorability.

Abstract

For integers $k, r > 0$, 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 at least $\min\{r, d(v)\}$ differently colored vertices. Given $r$, the smallest integer $k$ for which $G$ has a conditional $(k,r)$-coloring is called the $r$th order conditional chromatic number $\chi_r(G)$ of $G$. We give results (exact values or bounds for $\chi_r(G)$, depending on $r$) related to the conditional coloring of some graphs. We introduce \emph{unique conditional colorability} and give some related results. (Keywords. cartesian product of graphs; conditional chromatic number; gear graph; join of graphs.)