The $\Delta^2$ conjecture holds for graphs of small order

Abstract

An $L\left(2,1\right)$-labeling of a simple graph $G$ is a function $f:V\left(G\right)\to \mathbb{Z}$ such that if $xy\in E\left(G\right)$, then $|f\left(x\right)-f\left(y\right)|\ge 2$, and if the distance between $x$ and $y$ is two, then $|f\left(x\right)-f\left(y\right)|\ge 1$. $L\left(2,1\right)$-labelings are motivated by radio channel assignment problems. Denote by ${\lambda }_{2,1}\left(G\right)$ the smallest integer such that there exists an $L\left(2,1\right)$-labeling of $G$ using the integers $\left\{0,\dots ,{\lambda }_{2,1}\left(G\right)\right\}$. We prove that ${\lambda }_{2,1}\left(G\right)\le {\Delta }^2$, where $\Delta =\Delta \left(G\right)$, if the order of $G$ is no greater than $\left(\lfloor \Delta ∕2\rfloor +1\right)\left({\Delta }^2-\Delta +1\right)-1$. This shows that for graphs no larger than the given order, the 1992 “${\Delta }^2$ conjecture” of Griggs and Yeh holds. In fact, we prove more generally that if $L\ge {\Delta }^2+1$, $\Delta \ge 1$, and

$$\left|V\left(G\right)\right|\le \left(L-\Delta \right)\left(\lfloor \frac{L-1}{2\Delta }\rfloor +1\right)-1,$$

then ${\lambda }_{2,1}\left(G\right)\le L-1$. In addition, we exhibit an infinite family of graphs with ${\lambda }_{2,1}\left(G\right)={\Delta }^2-\Delta +1$.