On p-Intersection Representations

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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.

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Journal of Graph Theory