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arXiv · 1511.03913

We Found the Smallest Non-Autograph

Ben S. Baumer·Yijin Wei·Gary S. Bloom
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Abstract

Suppose that $G$ is a simple, vertex-labeled graph and that $S$ is a multiset. Then if there exists a one-to-one mapping between the elements of $S$ and the vertices of $G$, such that edges in $G$ exist if and only if the absolute difference of the corresponding vertex labels exist in $S$, then $G$ is an \emph{autograph}, and $S$ is a \emph{signature} for $G$. While it is known that many common families are graphs are autographs, and that infinitely many graphs are not autographs, a non-autograph has never been exhibited. In this paper, we identify the smallest non-autograph: a graph with 6 vertices and 11 edges. Furthermore, we demonstrate that the infinite family of graphs on $n$ vertices consisting of the complement of two non-intersecting cycles contains only non-autographs for $n \geq 8$.

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Ben S. Baumer, Yijin Wei, Gary S. Bloom. 2015-11-12. We Found the Smallest Non-Autograph. https://arxiv.org/abs/1511.03913

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