Cubicity, degeneracy, and crossing number

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 1 × R 2 × ⋯ × R 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. 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 ) log n ⌉ . In this paper we show the following: • k -degenerate graph G , cub ( G ) ≤ ( k + 2 ) ⌈ 2 e log n ⌉ . This bound is tight up to a constant factor. 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 ( n 2 k ) time to output an O ( k log n ) -dimensional cube representation for G . The above result has the following interesting consequences:• crossing number of a graph G is t , then box ( G ) is O ( t 1 4 ⌈ log t ⌉ 3 4 ) . This bound is tight up to a factor of O ( ( log t ) 1 4 ) . We also show that if G has n vertices, then cub ( G ) is O ( log n + t 1 / 4 log t ) . m ( P ) denote the poset dimension of a partially ordered set ( P , ≤ ) . We show that dim ( P ) ≤ 2 ( k + 2 ) ⌈ 2 e log n ⌉ , where k denotes the degeneracy of the underlying comparability graph of P . w that the cubicity of almost all graphs in the G ( n , m ) model is O ( d a v log n ) , where d a v = 2 m n denotes the average degree of the graph under consideration.