#### Title

On p-Intersection Representations

#### Document Type

Article

#### Date of Original Version

1-1-1996

#### Abstract

For a graph G = (V, E) and integer p, a p-intersection representation is a family ℱ = {Sx: x ∈ V} of subsets of a set S with the property that |Su ∩ Sv| ≥ p ⇔ {u, v} ∈ E. It is conjectured in [1] that θp(G) ≤ θ(Kn/2, n/2)(1 + o(1)) holds for any graPh with n vertices. This is known to be true for p = 1 by [4]. In [1], θ(Kn/2, n/2) ≥ (n2 + (2p - 1)n)/4p is proved for any n and p. Here, we show that this is asymptotically best possible. Further, we provide a bound on θp(G) for all graphs with bounded degree. In particular, we prove θp(G) ≤ O(n1/p) for any graph G with the maximum degree bounded by a constant. Finally, we also investigate the value of θp for trees. Improving on an earlier result of M. Jacobson, A. Kézdy, and D. West, (The 2-intersection number of paths and bounded-degree trees, preprint), we show that θ2(T) ≤ O(d√n) for any tree T with maximum degree dand θ2(T) ≤ O(n3/4) for any tree on n vertices. We conjecture that our result can be further improved and that θ2(T) ≤ O(√n) as long as Δ(T) ≤ √n. If this conjecture is true, our method gives θ2(T) ≤ O(n2/3) for any tree T which would be the best possible. © 1996 John Wiley & Sons, Inc.

#### Publication Title

Journal of Graph Theory

#### Volume

21

#### Issue

4

#### Citation/Publisher Attribution

Eaton, Nancy, Ronald J. Gould, and Vojtech Rödl.
"On p-Intersection Representations."
*Journal of Graph Theory*
21,
4
(1996): 377-392.
doi:10.1002/(sici)1097-0118(199604)21:4<377::aid-jgt3>3.3.co;2-2.