Tight Upper Bound on the Clique Size in the Square of 2-Degenerate Graphs

The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most 2. In general, $\Delta \left(G\right)+1\le \chi \left(G^2\right)\le \Delta {\left(G\right)}^2+1$ for every graph $G$. Charpentier (2014) asked whether $\chi \left(G^2\right)\le 2\Delta \left(G\right)$ if $\mathrm{mad}\left(G\right)<4$. But Hocquard, Kim, and Pierron (2019) answered his question negatively. For every even value of $\Delta \left(G\right)$g, they constructed a 2-degenerate graph $G$ such that $\omega \left(G^2\right)=\frac{5}{2}\Delta \left(G\right)$. Note that if $G$ is a 2-degenerate graph, then $mad\left(G\right)<4$. Thus, we have that

$\frac{5}{2}\Delta \left(G\right)\le \max \{\chi \left(G^2\right):G\text{is a 2-degenerate graph}\}\le 3\Delta \left(G\right)+1.$

So, it was naturally asked whether there exists a constant $D_0$ such that $\chi \left(G^2\right)\le \frac{5}{2}\Delta \left(G\right)$ if $G$ is a 2-degenerate graph with $\Delta \left(G\right)\ge D_0$. Recently, Cranston and Yu (2024) showed that $\omega \left(G^2\right)\le \frac{5}{2}\Delta \left(G\right)+72$ if $G$ is a 2-degenerate graph, and $\omega \left(G^2\right)\le \frac{5}{2}\Delta \left(G\right)+60$ if $G$ is a 2-degenerate graph with $\Delta \left(G\right)\ge 1729$. We show that there exists a constant $D_0$ such that $\omega \left(G^2\right)\le \frac{5}{2}\Delta \left(G\right)$ if $G$ is a 2-degenerate graph with $\Delta \left(G\right)\ge D_0$. This upper bound on $\omega \left(G^2\right)$ is tight by the construction in Hocquard, Kim, and Pierron.