On conditional coloring of some graphs

TL;DR

This paper investigates the conditional chromatic number of various graph classes, including grids, squares of cycles, strong products, and web graphs, providing specific values for different parameters.

Contribution

It determines the conditional chromatic number for several graph families under various conditions, extending existing coloring concepts.

Findings

Conditional chromatic number of grid graphs is established.

Results include the second order conditional chromatic number of (t,n)-web graphs.

Provides formulas for the conditional chromatic number of strong product graphs.

Abstract

For integers r and k > 0(k>r),a conditional (k, r)-coloring of a graph G is a proper k-coloring of G such that every vertex v of G has at least min{r,d(v)} differently colored neighbors, where d(v) is the degree of v. In this note, for different values of r we obtain the conditional chromatic number of a grid $G(2,n) \cong P_2 \ \Box \ P_n$, $C_n^2$ and the strong product of $P_n$ and $P_m$ (n,m being positive integers). Also, for integers $n \geq 3$ and $t \geq 1$ the second order conditional chromatic number (also known as dynamic chromatic number) of the (t,n)-web graph is obtained.