#### Title

Set Intersection Representations for Almost All Graphs

#### Document Type

Article

#### Date of Original Version

1-1-1996

#### Abstract

Two variations of set intersection representation are investigated and upper and lower bounds on the minimum number of labels with which a graph may be represented are found that hold for almost all graphs. Specifically, if θk(G) is defined to be the minimum number of labels with which G may be represented using the rule that two vertices are adjacent if and only if they share at least k labels, there exist positive constants ck and c′k such that almost every graph G on n vertices satisfies ckn2/log2n ≤ θk(G) ≤ c′kn2/log2n. Changing the representation only slightly by defining θodd(G) to be the minimum number of labels with which G can be represented using the rule that two vertices are adjacent if and only if they share an odd number of labels results in quite different behavior. Namely, almost every graph G satisfies n - √2n - [log n] < θodd(G) ≤ n - 1.

#### Publication Title, e.g., Journal

Journal of Graph Theory

#### Volume

23

#### Issue

3

#### Citation/Publisher Attribution

Eaton, Nancy, and David A. Grable.
"Set Intersection Representations for Almost All Graphs."
*Journal of Graph Theory*
23,
3
(1996): 309-320.
doi: 10.1002/(SICI)1097-0118(199611)23:3<309::AID-JGT11>3.0.CO;2-9.