On large bipartite graphs of diameter 3

TL;DR

This paper investigates the maximum size of bipartite graphs with given degree and diameter, proving non-existence or uniqueness of certain extremal graphs, and discovering new large graphs that improve known bounds.

Contribution

It provides structural insights into bipartite graphs of diameter 3, proves non-existence of specific extremal graphs, and identifies new large graphs that set improved bounds.

Findings

No bipartite (7,3,-4)-graphs exist, confirming the optimality of the (7,3,-6)-graph.

The bipartite (5,3,-4)-graph is unique.

Discovered three new bipartite vertex-transitive graphs of degree 11, diameter 3, order 190.

Abstract

We consider the bipartite version of the {\it degree/diameter problem}, namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number $\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter $D$. In this context, the bipartite Moore bound $\M^b(d,D)$ represents a general upper bound for $\N^b(d,D)$. Bipartite graphs of order $\M^b(d,D)$ are very rare, and determining $\N^b(d,D)$ still remains an open problem for most $(d,D)$ pairs. This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on bipartite $(d,D,-4)$-graphs (that is, bipartite graphs of order $\M^b(d,D)-4$) was carried out. Here we first present some structural properties of bipartite $(d,3,-4)$-graphs, and later prove there are no bipartite $(7,3,-4)$-graphs. This result implies that the known bipartite $(7,3,-6)$-graph is optimal, and therefore $\N^b(7,3)=80$. Our approach also bears a proof of the uniqueness of the known bipartite $(5,3,-4)$-graph, and the non-existence of bipartite $(6,3,-4)$-graphs. In addition, we discover three new largest known bipartite (and also vertex-transitive) graphs of degree 11, diameter 3 and order 190, result which improves by 4 vertices the previous lower bound for $\N^b(11,3)$.