On the cubicity of bipartite graphs

TL;DR

This paper investigates the cubicity of bipartite graphs, providing upper bounds based on degrees and sizes, and introduces an efficient randomized algorithm for constructing low-dimensional cube representations.

Contribution

It establishes new upper bounds for bipartite graph cubicity using degree and size parameters, and presents an efficient randomized algorithm for cube representation construction.

Findings

Upper bound: cub(G) ≤ 2(Δ'+2) ⌈ln n₂⌉ for bipartite graphs.

Efficient randomized algorithm for cube representation in 3(Δ'+2) ⌈ln n₂⌉ dimensions.

Potential for improved graph problem solutions using low-dimensional cube representations.

Abstract

{\it A unit cube in $k$-dimension (or a $k$-cube) is defined as the cartesian product $R_1 \times R_2 \times ... \times R_k$, where each $R_i$ is a closed interval on the real line of the form $[a_i, a_i+1]$. The {\it cubicity} of $G$, denoted as $cub(G)$, is the minimum $k$ such that $G$ is the intersection graph of a collection of $k$-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph. It is known that for a graph $G$, $cub(G) \leq \lfloor\frac{2n}{3}\rfloor$. Recently it has been shown that for a graph $G$, $cub(G) \leq 4(\Delta + 1)\ln n$, where $n$ and $\Delta$ are the number of vertices and maximum degree of $G$, respectively. In this paper, we show that for a bipartite graph $G = (A \cup B, E)$ with $|A| = n_1$, $|B| = n_2$, $n_1 \leq n_2$, and $\Delta' = \min\{\Delta_A, \Delta_B\}$, where $\Delta_A = {max}_{a \in A}d(a)$ and $\Delta_B = {max}_{b \in B}d(b)$, $d(a)$ and $d(b)$ being the degree of $a$ and $b$ in $G$ respectively, $cub(G) \leq 2(\Delta'+2) \lceil \ln n_2 \rceil$. We also give an efficient randomized algorithm to construct the cube representation of $G$ in $3(\Delta'+2)\lceil \ln n_2 \rceil$ dimensions. The reader may note that in general $\Delta'$ can be much smaller than $\Delta$.}