Packing coloring of some undirected and oriented coronae graphs

TL;DR

This paper investigates the packing chromatic number of generalized corona graphs of paths and cycles, extending the concept to directed graphs and determining their packing chromatic numbers.

Contribution

It provides the first comprehensive analysis of packing chromatic numbers for generalized coronae of paths and cycles, including their orientations and directed variants.

Findings

Determined packing chromatic numbers for coronae of paths and cycles.

Extended packing coloring to directed graphs with orientation considerations.

Provided exact values for various classes of corona graphs.

Abstract

The packing chromatic number $\pcn(G)$ of a graph $G$ is the smallest integer $k$ such that its set of vertices $V(G)$ can be partitioned into $k$ disjoint subsets $V\_1$, \ldots, $V\_k$, in such a way that every two distinct vertices in $V\_i$ are at distance greater than $i$ in $G$ for every $i$, $1\le i\le k$. For a given integer $p \ge 1$, the generalized corona $G\odot pK\_1$ of a graph $G$ is the graph obtained from $G$ by adding $p$ degree-one neighbors to every vertex of $G$. In this paper, we determine the packing chromatic number of generalized coronae of paths and cycles. Moreover, by considering digraphs and the (weak) directed distance between vertices, we get a natural extension of the notion of packing coloring to digraphs. We then determine the packing chromatic number of orientations of generalized coronae of paths and cycles.