Injective (edge) colorings of generalized Sierpi\'{n}ski graphs

Abstract

Generalized Sierpi\'{n}ski graphs constitute a distinctive class of fractal-like networks with recursive definition: given a graph $G$, $S_G^1=G$ while $S_G^n$ is obtained from $|V(G)|$ copies of $S_G^{n-1}$ by adding some edges in a prescribed way that reflects the structure of $G$. Many graph invariants have been studied in generalized Sierpi\'{n}ski graphs. In this paper, we focus on their injective colorings, both the vertex and the edge version. Given a graph $G$, a mapping $f$ that assigns an integer from $\{1,\ldots,k\}$ to each vertex (resp.\ edge) of $G$ is an injective (edge) coloring of $G$ if $f(x)=f(y)$ implies that $x$ and $y$ are not in a common triangle nor at distance $2$ for any two vertices (resp.\ edges) $x$ and $y$ in $G$. The minimum number of colors $k$ for which there exists an injective (edge) coloring of $G$ is called the injective chromatic number (resp.\ injective chromatic index) of $G$ and is denoted by $\chi_i(G)$ (resp.\ $\chi_i'(G)$). The vertex version of injective colorings in generalized Sierpi\'{n}ski graphs was studied in an earlier paper, where the authors determined the injective chromatic numbers of standard Sierpi\'{n}ski graphs, and asked about the values when $G$ is a cycle. We resolve this question by proving that $\chi_i(S_{C_k}^n)=3$ for every $n\ge 2$ and every $k\ge 3$. Moreover, we prove an almost conclusive result that $\chi_i(S_G^n)\in \{\chi_i(G),\chi_i(G)+1\}$ for any graph $G$ and any $n\ge 2$. For injective edge colorings we prove that $\chi_i'(S_{K_3}^n)=5$ for all $n\ge 3$, while $\chi_i'(S_{K_3}^2)=4$ and $\chi_i'(S_{K_3}^1)=3$. Furthermore, if $G$ is a triangle-free graph, we prove that $\chi_i'(S_G^n)\in \{\chi_i'(S_G^3),\chi_i'(S_G^3)+1\}$ for all $n\ge 4$, and provide some sufficient conditions on an injective edge coloring of the 3-dimensional Sierpi\'{n}ski graph over $G$, which ensure that $\chi_i'(S_G^n)=\chi_i'(S_G^3)$.