On the Conjecture of the Representation Number of Bipartite Graphs

TL;DR

This paper investigates the representation number of bipartite graphs, providing new bounds and partial proofs for a conjecture relating the representation number to the sizes of the partite sets.

Contribution

The paper establishes new upper bounds on the representation number of bipartite graphs based on the smallest partite set size and confirms the conjecture for certain subclasses.

Findings

Every bipartite graph is (1 + ⌈m/2⌉)-representable.

If m is odd, bipartite graphs are ⌈m/2⌉-representable.

The conjecture holds for bipartite graphs with unequal partite set sizes, except for equal and even sizes.

Abstract

While the problem of determining the representation number of an arbitrary word-representable graph is NP-hard, this problem is open even for bipartite graphs. The representation numbers are known for certain bipartite graphs including all the graphs with at most nine vertices. For bipartite graphs with partite sets of sizes $m$ and $n$, Glen et al. conjectured that the representation number is at most $\lceil \frac{m+n}{4}\rceil$, where $m+n \ge 9$. In this paper, we show that every bipartite graph is $\left( 1+ \lceil \frac{m}{2} \rceil \right)$-representable, where $m$ is the size of its smallest partite set. Furthermore, if $m$ is odd then we prove that the bipartite graphs are $\lceil \frac{m}{2} \rceil $-representable. Accordingly, we establish that the conjecture by Glen et al. holds good for all bipartite graphs leaving the bipartite graphs whose partite sets are of equal and even size. In case of the bipartite graphs with partite sets of equal and even size, we prove the conjecture for certain subclasses using the neighborhood inclusion graph approach.