On Sierpi\'{n}ski packing chromatic number and recognition of Sierpi\'{n}ski products

TL;DR

This paper investigates the Sierpiński packing chromatic number of specific graph products, providing exact values for complete graphs and bounds for paths and stars, and offers a polynomial-time recognition method for certain Sierpiński product graphs.

Contribution

It determines the (upper) Sierpiński packing chromatic number for complete, path, and star graph factors, and introduces a polynomial-time recognition algorithm for Sierpiński product graphs with tree factors.

Findings

Exact Sierpiński packing chromatic number for complete graph factors.

Bounded upper Sierpiński packing chromatic number for paths and stars.

Polynomial-time recognition of Sierpiński product graphs with tree factors.

Abstract

The Sierpi\'{n}ski product $G \otimes _f H$ of graphs $G$ and $H$ with respect to a function $f \colon V(G)\rightarrow V(H)$ has the vertex set $V(G)\times V(H)$. For every $g\in V(G)$ it contains a disjoint copy $gH$ of $H$, and for every edge $gg'$ of $G$ there is the edge $(g,f(g'))(g',f(g))$ between $gH$ and $g'H$. In this paper, the Sierpi\'{n}ski packing chromatic number is defined as the minimum of $\chi_{\rho}(G\otimes _f H)$ over all functions $f$, where $\chi_{\rho}(X)$ is the packing chromatic number of $X$. The upper Sierpi\'{n}ski packing chromatic number is analogously defined as the maximum corresponding value. The (upper) Sierpi\'{n}ski packing chromatic number is determined for all Sierpi\'{n}ski product graphs whose both factors are complete. Sierpi\'{n}ski product graphs whose factors are paths or stars are also studied. Their Sierpi\'{n}ski packing chromatic number is always $3$, while their upper Sierpi\'{n}ski packing chromatic number is bounded from below and above. It is also proved that for a given graph $G$, it can be checked in polynomial time whether $G$ has a representation as a Sierpi\'{n}ski product graphs both factors of which being trees.