A Classification of Hyperfocused 12-Arcs

Abstract

A $k$-arc in PG($2,q$) is a set of $k$ points no three of which are collinear. A hyperfocused $k$-arc is a $k$-arc in which the $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ secants meet some external line in exactly $k-1$ points. Hyperfocused $k$-arcs can be viewed as 1-factorizations of the complete graph $K_k$ that embed in PG($2,q$). We study the 526,915,620 1-factorizations of $K_{12}$, determine which are embeddable in PG($2,q$), and classify hyperfocused $12$-arcs. Specifically we show if a $12$-arc $\mathcal{K}$ is a hyperfocused arc in PG($2,q$) then $q=2^{5k}$ and $\mathcal{K}$ is a subset of a hyperconic including the nucleus.