Cubicity, degeneracy, and crossing number

• Authors: Abhijin Adiga;L. Sunil Chandran;Rogers Mathew
• Venue: European Journal of Combinatorics
• Year: 2014

A k-box B=(R"1,...,R"k), where each R"i is a closed interval on the real line, is defined to be the Cartesian product R"1xR"2x...xR"k. If each R"i is a unit-length interval, we call B a k-cube. The 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, Naveen Sivadasan. Cubicity and bandwidth. Graphs and Combinatorics 29 (1) (2013) 45-69] that, for a graph G with n vertices and maximum degree @D, cub(G)@?@?4(@D+1)logn@?. In this paper we show the following: *For a k-degenerate graph G, cub(G)@?(k+2)@?2elogn@?. This bound is tight up to a constant factor. Since k is at most @D and can be much lower, this clearly is an asymptotically stronger result. Moreover, we have an efficient deterministic algorithm that runs in O(n^2k) time to output an O(klogn)-dimensional cube representation for G. The above result has the following interesting consequences:*If the crossing number of a graph G is t, then box(G) is O(t^1^4@?logt@?^3^4). This bound is tight up to a factor of O((logt)^1^4). We also show that if G has n vertices, then cub(G) is O(logn+t^1^/^4logt). *Let dim(P) denote the poset dimension of a partially ordered set (P,@?). We show that dim(P)@?2(k+2)@?2elogn@?, where k denotes the degeneracy of the underlying comparability graph of P. *We show that the cubicity of almost all graphs in the G(n,m) model is O(d"a"vlogn), where d"a"v=2mn denotes the average degree of the graph under consideration.