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arXiv · 1711.03856

Packing coloring of Sierpiński-type graphs

Boštjan Brešar·Jasmina Ferme
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Abstract

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

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Boštjan Brešar, Jasmina Ferme. 2017-11-10. Packing coloring of Sierpiński-type graphs. https://arxiv.org/abs/1711.03856

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