Distance Graphs on the Integers

We consider several extremal problems concerning representations of graphs as distance graphs on the integers. Given a graph $G=(V,E)$, we wish to find an injective function $\phi:V\to{\mathbb Z}^+=\{1,2,\dots\}$ and a set ${\mathcal D}\subset{\mathbb Z}^+$ such that $\{u,v\}\in E$ if and only if $|\phi(u)-\phi(v)|\in{\mathcal D}$.Let $s(n)$ be the smallest $N$ such that any graph $G$ on $n$ vertices admits a representation $(\phi_G,{\mathcal D}_G)$ such that $\phi_G(v)\leq N$ for all $v\in V(G)$. We show that $s(n)=(1+o(1))n^2$ as $n\to\infty$. In fact, if we let $s_r(n)$ be the smallest $N$ such that any $r$-regular graph $G$ on $n$ vertices admits a representation $(\phi_G,{\mathcal D}_G)$ such that $\phi_G(v)\leq N$ for all $v\in V(G)$, then $s_r(n)=(1+o(1))n^2$ as $n\to\infty$ for any $r=r(n)\gg\log n$ with $rn$ even for all $n$.Given a graph $G=(V,E)$, let $D_{\rm e}(G)$ be the smallest possible cardinality of a set ${\mathcal D}$ for which there is some $\phi\:V\to{\mathbb Z}^+$ so that $(\phi,{\mathcal D})$ represents $G$. We show that, for almost all $n$-vertex graphs $G$, we have \begin{equation*} D_{\rm e}(G)\geq\frac{1}{2}\binom{n}{2}-(1+o(1))n^{3/2}(\log n)^{1/2}, \end{equation*} whereas for some $n$-vertex graph $G$, we have \begin{equation*} D_{\rm e}(G)\geq\binom{n}{2}-n^{3/2}(\log n)^{1/2+o(1)}.\end{equation*} Further extremal problems of similar nature are considered.