Packing chromatic numbers of finite super subdivisions of graphs

TL;DR

This paper investigates the packing chromatic number of finite super subdivisions of graphs, providing general properties and exact values for specific graph classes such as complete graphs, cycles, and neighborhood corona graphs.

Contribution

It introduces new results on the packing chromatic number for finite super subdivisions, including exact calculations for several important graph families.

Findings

Determined packing chromatic numbers for super subdivisions of complete graphs and cycles.

Established general properties of packing chromatic numbers in super subdivisions.

Provided exact values for neighborhood corona graphs of cycles and paths.

Abstract

The \textit{packing chromatic number} of a graph $G$, denoted by $% \chi_\rho(G)$, is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In this paper, we present some general properties of packing chromatic numbers of \textit{finite super subdivisions} of graphs. We determine the packing chromatic numbers of the finite super subdivisions of complete graphs, cycles and \textit{neighborhood corona graphs} of a cycle and a path respectively of a complete graph and a path.