Graphs of small dimensions

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Let G=(V, E) be a graph with n vertices. The direct product dimension pdim(G) (c.f. [10], [12]) is the minimum number t such that G can be embedded into a product of t copies of complete graphs Kn. In [10], Lovász, Nešetřil and Pultr determined the direct product dimension of matchings and paths and gave sharp bounds for the product dimension of cycles, all logarithmic in the number of vertices. Here we prove that pdim(G)≤cd log n for any graph with maximum degree d and n vertices and show that up to a factor of 1/ (log d + log log n/2d) this bound is the best possible. We also study set representations of graphs. Let G=(V,E) be a graph and p≥1 an integer. A family script J sign={Ax, x ∈ V} of (not necessarily distinct) sets is called a p-intersection representation of G if |Ax ∩ Ay| ≥ p ⇔ {x,y} ∈ E for every pair x, y of distinct vertices of G. Let θp(G) be the minimum size of \Uscript J sign\ taken over all intersection representations of G. We also study the parameter θ(G)=minp(θp(G)). It turns out that these parameters can be bounded in terms of maximum degree and linear density of a graph G or its complement Ḡ. While for example,θ1(G)=\E(G)\ holds if G contains no triangle, N. Alon proved that θ1(G)≤cΔ(Ḡ) logn, where Δ(Ḡ) denotes the maximum degree of Ḡ. We extend this by showing that Δ(Ḡ) can be replaced by ρ(Ḡ), the linear density of Ḡ. We also show that this bound is close to best possible as there are graphs with θ1(G) ≥ c2 Δ2(Ḡ)/logΔ(G) log n. For the parameter θ we conjecture that θ(G) ≤ cΔ(Ḡ)1+ε log n for some constant c not dependent on Δ(Ḡ) or n and show that θ(G) ≤ cΔ(Ḡ) log2 Δ(Ḡ) log n if G is bipartite. This is, up to the factor 1/log2 Δ(Ḡ) best possible. Finally we give an upper bound on θ(G) in terms of Δ(G) and prove θ(G) ≤ cΔ2(G) log n.

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