Packing coloring of Sierpi\'{n}ski-type graphs

TL;DR

This paper investigates the packing chromatic number of various Sierpiński-type graphs, revealing bounds and unboundedness depending on the graph's base and structure, advancing understanding of graph coloring properties.

Contribution

It establishes bounds and unboundedness results for the packing chromatic number in different families of Sierpiński graphs, including generalized cases.

Findings

Packing chromatic number is bounded by 8 for Sierpiński graphs with base 3.

It is unbounded for bases greater than 3.

Upper bound of 31 for Sierpiński triangle graphs.

Abstract

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ 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 consider the packing chromatic number of several families of Sierpi\'{n}ski-type graphs. While it is known that this number is bounded from above by $8$ in the family of Sierpi\'{n}ski graphs with base $3$, we prove that it is unbounded in the families of Sierpi\'{n}ski graphs with bases greater than $3$. On the other hand, we prove that the packing chromatic number in the family of Sierpi\'{n}ski triangle graphs $ST^n_3$ is bounded from above by $31$. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpi\'{n}ski graphs $S^n_G$ with respect to all connected graphs $G$ of order 4.