Graphs of small dimensions

Abstract

LetG=(V, E) be a graph withn vertices. The direct product dimension pdim (G) (c.f. [10], [12]) is the minimum numbert such thatG can be embedded into a product oft copies of complete graphsK n. In [10], Lovasz, Nesetřil 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)≤cdlogn for any graph with maximum degreed andn vertices and show that up to a factor of $$1/\left( {\log d + \log \log \frac{n}{{2d}}} \right)$$ this bound is the best possible. We also study set representations of graphs. LetG=(V,E) be a graph andp≥1 an integer. A familyF={A x ,x∈V} of (not necessarily distinct) sets is called ap-intersection representation ofG if |A x↓A y|≥p⇔{x,y}∈E for every pairx, y of distinct vertices ofG. Let θ G (G) be the minimum size of |U F| taken over all intersection representations ofG. We also study the parameter θ(G)= $$\mathop {\min }\limits_p $$ (θ p (G)). It turns out that these parameters can be bounded in terms of maximum degree and linear density of a graphG or its complement $$\bar G$$ . While for example, θ1(G)=|E(G)| holds ifG contains no triangle, N. Alon proved that θ1(G)≤cΔ( $$\bar G$$ ) logn, where Δ( $$\bar G$$ ) denotes the maximum degree of $$\bar G$$ . We extend this by showing that Δ( $$\bar G$$ ) can be replaced by ϱ( $$\bar G$$ ), the linear density of $$\bar G$$ . We also show that this bound is close to best possible as there are graphs with $$\theta _1 (G) \geqslant c_2 \frac{{\Delta ^2 (\bar G)}}{{\log \Delta (G)}}$$ logn. For the parameter θ we conjecture that $$\theta (G) \leqslant c\Delta (\bar G)^{1 + \varepsilon } \log n$$ for some constantc not dependent on Δ( $$\bar G$$ ) orn and show that θ(G)≤cΔ( $$\bar G$$ ) log2 Δ( $$\bar G$$ ) logn ifG is bipartite. This is, up to the factor 1/log2Δ( $$\bar G$$ ) best possible. Finally, we give an upper bound on θ(G) in terms of Δ(G) and prove θ(G)≤cΔ2(G) logn.