Injective colorings of Sierpi\'nski-like graphs and Kneser graphs

TL;DR

This paper explores the injective chromatic number of Sierpiński-like and Kneser graphs, establishing relationships with other graph invariants and identifying conditions for perfect injective colorability.

Contribution

It introduces new bounds and properties of the injective chromatic number for Sierpiński and Kneser graphs, including their perfect injective colorability.

Findings

Sierpiński graphs are Class 1.

Injective chromatic number of rooted products is within six possible values.

Kneser graphs $K(n,r)$ with $n \\ge 3r-1$ are perfect injectively colorable.

Abstract

Two relationships between the injective chromatic number and, respectively, chromatic number and chromatic index, are proved. They are applied to determine the injective chromatic number of Sierpi\'nski graphs and to give a short proof that Sierpi\'nski graphs are Class $1$. Sierpi\'nski-like graphs are also considered, including generalized Sierpi\'nski graphs over cycles and rooted products. It is proved that the injective chromatic number of a rooted product of two graphs lies in a set of six possible values. Sierpi\'nski graphs and Kneser graphs $K(n,r)$ are considered with respect of being perfect injectively colorable, where a graph is perfect injectively colorable if it has an injective coloring in which every color class forms an open packing of largest cardinality. In particular, all Sierpi\'nski graphs and Kneser graphs $K(n, r)$ with $n \ge 3r-1$ are perfect injectively colorable graph, while $K(7,3)$ is not.