We Found the Smallest Non-Autograph

TL;DR

This paper identifies the smallest graph that cannot be represented as an autograph, a graph defined by a specific vertex-labeling and difference-based edge condition, and explores a family of non-autograph graphs.

Contribution

The paper proves the existence of the smallest non-autograph with 6 vertices and 11 edges, and characterizes an infinite family of non-autograph graphs for larger sizes.

Findings

Smallest non-autograph has 6 vertices and 11 edges.

Infinite family of non-autograph graphs for n ≥ 8.

Many common graphs are autographs, but some are not.

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 $n \geq 8$.