Cubicity, degeneracy, and crossing number

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Abstract

A k-box B=(R1,,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. 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 Δ, cub(G)4(Δ+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 Δ and can be much lower, this clearly is an asymptotically stronger result. Moreover, we have an efficient deterministic algorithm that runs in O(n2k) 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(t14logt34). This bound is tight up to a factor of O((logt)14). We also show that if G has n vertices, then cub(G) is O(logn+t1/4logt).

  • 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(davlogn), where dav=2mn denotes the average degree of the graph under consideration.

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