#### Document Type

Article

#### Publication Date

2016

#### Publication Title

Discussiones Mathematicae Graph Theory

#### 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 autograph, and S is a signature for G. While it is known that many common families of 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≥8.

#### Keywords

graph labeling, difference graphs, autographs, monographs

#### Volume

36

#### First Page

577

#### Last Page

602

#### DOI

doi:10.7151/dmgt.1881

#### Rights

Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.

#### Recommended Citation

Baumer, Benjamin; Wei, Yijin; and Bloom, Gary S., "The Smallest Non-Autograph" (2016). *Mathematics and Statistics: Faculty Publications*. 18.

http://scholarworks.smith.edu/mth_facpubs/18