Cubicity, Degeneracy, and Crossing Number

A k-box B = (R1, R2, . . . , Rk), where each Ri is a closed interval on the real line, is defined to be the Cartesian product R1 × R2 × · · · × Rk. If each Ri is a unit length interval, we call B a k-cube. Boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k such that G is an intersection graph of k-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan. Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/0607092, 2006.] that, for a graph G with maximum degree ∆, cub(G) ≤ d4(∆ + 1) lnne. In this paper we show that, for a k-degenerate graph G, cub(G) ≤ (k+2)d2e logne. Since k is at most ∆ and can be much lower, this clearly is a stronger result. We also give an efficient deterministic algorithm that runs in O(n2k) time to output a 8k(d2.42 logne+ 1) dimensional cube representation for G. The crossing number of a graph G, denoted as CR(G), is the minimum number of crossing pairs of edges, over all drawings of G in the plane. An important consequence of the above result is that if the crossing number of a graph G is t, then box(G) is O(t1/4dlog te) . This bound is tight upto a factor of O((log t)3/4). Let (P,≤) be a partially ordered set and let GP denote its underlying comparability graph. Let dim(P) denote the poset dimension of P. Another interesting consequence of our result is to show that dim(P) ≤ 2(k + 2)d2e logne, where k denotes the degeneracy of GP . Also, we get a deterministic algorithm that runs in O(n2k) time to construct a 16k(d2.42 logne + 1) sized realizer for P. As far as we know, though very good upper bounds exist for poset dimension in terms of maximum degree of its underlying comparability graph, no upper bounds in terms of the degeneracy of the underlying comparability graph is seen in the literature. 1998 ACM Subject Classification G.2.2 Graph Theory