On the Conjecture of the Representation Number of Bipartite Graphs

2025-06-01

AI Summary

This paper investigates the upper bound on the representation number of bipartite graphs, aiming to verify Glen et al.’s conjecture that every bipartite graph with bipartition sizes (m) and (n) (where (m+n geq 9)) has representation number at most (lceil (m+n)/4 ceil). Addressing the open case of balanced even-sized bipartitions, we establish, for the first time, a unified upper bound parameterized by the smaller part size (m): all bipartite graphs are ((1 + lceil m/2 ceil))-representable, and (lceil m/2 ceil)-representability is achievable when (m) is odd. Our approach combines neighborhood-inclusion graph constructions with case-based induction. We fully resolve the conjecture for all bipartite graphs with odd-sized smaller parts and substantially advance its verification for balanced even-sized bipartitions. The results confirm Glen’s conjecture for all bipartite graphs except possibly some balanced even-sized ones, and we explicitly verify it for multiple representative classes of such graphs. This significantly improves both the precision and scope of existing upper bounds.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Uncertainty Representations
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 $lceil frac{m+n}{4} ceil$, where $m+n ge 9$. In this paper, we show that every bipartite graph is $left( 1+ lceil frac{m}{2} ceil ight)$-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} ceil $-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.

Problem

• Determining representation number for bipartite graphs
• Verifying Glen et al.'s conjecture on representation bounds
• Handling bipartite graphs with equal even-sized partite sets

Innovation

• Bipartite graphs representation number bounds
• Neighborhood inclusion graph approach
• Odd smallest partite set representation