Product Dimension of Forests and Bounded Treewidth Graphs

TL;DR

This paper establishes new upper bounds on the product dimension for forests, bounded treewidth graphs, and k-degenerate graphs, improving previous results through novel techniques involving orthogonal Latin squares.

Contribution

It provides tighter bounds on the product dimension for specific graph classes, extending the understanding of graph product representations.

Findings

Forests have a product dimension at most 1.441logn+3.

Graphs with treewidth t have a product dimension at most (t+2)(logn+1).

k-degenerate graphs have a product dimension at most loor{8.317klogn}+1.

Abstract

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has a product dimension at most 1.441logn+3. This improves the best known upper bound of 3logn for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a result on existence of orthogonal Latin squares to show that every graph on n vertices with a treewidth at most t has a product dimension at most (t+2)(logn+1). We also show that every k-degenerate graph on n vertices has a product dimension at most \ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by Eaton and Rodl.