Boxicity of Line Graphs

TL;DR

This paper investigates the boxicity of line graphs and hypercubes, providing upper bounds for line graphs of multigraphs and lower bounds for hypercubes, advancing understanding of geometric representations of these graphs.

Contribution

It establishes new bounds on the boxicity of line graphs of multigraphs and hypercubes, and explores the boxicity of fully subdivided graphs, addressing open questions in the field.

Findings

Upper bound: box(G) <= 2Δ(ceil(log2(log2(Δ))) + 3) + 1 for line graphs of multigraphs

Lower bound: box(H_d) >= (ceil(log2(log2(d))) + 1)/2 for hypercubes

Provides bounds for boxicity of fully subdivided graphs

Abstract

Boxicity of a graph H, denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this paper, we show that for a line graph G of a multigraph, box(G) <= 2\Delta(\lceil log_2(log_2(\Delta)) \rceil + 3) + 1, where \Delta denotes the maximum degree of G. Since \Delta <= 2(\chi - 1), for any line graph G with chromatic number \chi, box(G) = O(\chi log_2(log_2(\chi))). For the d-dimensional hypercube H_d, we prove that box(H_d) >= (\lceil log_2(log_2(d)) \rceil + 1)/2. The question of finding a non-trivial lower bound for box(H_d) was left open by Chandran and Sivadasan in [L. Sunil Chandran and Naveen Sivadasan. The cubicity of Hypercube Graphs. Discrete Mathematics, 308(23):5795-5800, 2008]. The above results are consequences of bounds that we obtain for the boxicity of fully subdivided graphs (a graph which can be obtained by subdividing every edge of a graph exactly once).