On the Representation Number of Bipartite Graphs

TL;DR

This paper introduces a polynomial-time relabeling algorithm that constructs a word representation for bipartite graphs as concatenations of permutations, providing an upper bound on their representation number and related poset dimension.

Contribution

It presents the first polynomial-time algorithm to produce such word representations for bipartite graphs, establishing an upper bound on their representation number.

Findings

Algorithm runs in polynomial time.

Provides an upper bound for bipartite graph representation number.

Links bipartite graph representations to poset dimension.

Abstract

A word-representable graph is a simple graph $G$ which can be represented by a word $w$ over the vertices of $G$ such that any two vertices are adjacent in $G$ if and only if they alternate in $w$. It is known that the class of comparability graphs -- the graphs which admit a transitive orientation -- is precisely the class of graphs that can be represented by a concatenation of permutations of vertices. The class of bipartite graphs is a subclass of comparability graphs. While it is an open problem to determine the representation number of comparability graphs, it was conjectured that the representation number of bipartite graphs on $n$ vertices is at most $n/4$. In this paper, we propose a polynomial time relabeling algorithm to produce a word representing a given bipartite graph which is a concatenation of permutations of the graph's vertices. Thus we obtain an upper bound for the representation number of bipartite graphs, which in turn gives us an upper bound for the dimension of the posets corresponding to bipartite graphs.