Conditional coloring of some parameterized graphs

TL;DR

This paper investigates the conditional coloring properties of various parameterized graphs, introducing the concept of the rth order conditional chromatic number and calculating it for specific graph classes.

Contribution

It defines the rth order conditional chromatic number and computes it for several classes of parameterized graphs, expanding understanding of graph coloring constraints.

Findings

Determined $ ext{χ}_r(G)$ for Windmill graphs.

Calculated $ ext{χ}_r(G)$ for line graphs of Windmill graphs.

Analyzed middle graphs of various graph classes.

Abstract

For integers k>0 and 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 vertices with at least min{r,d(v)} different colors. 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 $\chi_r(G)$. For different values of r 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 and middle graph of a bipartite graph.