Intersection representation of complete unbalanced bipartite graphs
Document Type
Article
Date of Original Version
11-1-1997
Abstract
Ap-intersectionrepresentation of a graphGis a map,f, that assigns each vertex a subset of {1,2,...,t} such that {u,v} is an edge if and only if |f(u)∩f(v)|≥p. The symbolθp(G) denotes this minimumtsuch that ap-intersection representation ofGexists. In 1966 Erdos, Goodman, and Pósa showed that for all graphsGon 2nvertices,θ1(G)≤θ1(Kn,n)=n 2. In 1992, Chung and West conjectured that for all graphsGon 2nvertices,θp(G)≤θp(Kn,n) whenp≥1. Subsequently, upper and lower bounds forθp(Kn,n) have been found to be (n2/p)(1+o(1)). We show in this paper that many complete unbalanced bipartite graphs on 2nvertices have a largerp-intersection number thanKn,n. For example, whenp=2,θ2(Kn,n)≤12n2(1+o(1))<4172n 2(1+o(1))≤θ2(K(5/6)n,(7/6)n). © 1997 Academic Press.
Publication Title, e.g., Journal
Journal of Combinatorial Theory. Series B
Volume
71
Issue
2
Citation/Publisher Attribution
Eaton, Nancy. "Intersection representation of complete unbalanced bipartite graphs." Journal of Combinatorial Theory. Series B 71, 2 (1997): 123-129. doi: 10.1006/jctb.1997.1779.