Size of bipartite graphs with given diameter and connectivity constraints

Abstract

In the first part of this paper we determine the maximum size of a (finite, simple, connected) bipartite graph of given order, diameter $d$, and connectivity $\kappa$. It was shown by Ali, Mazorodze, Mukwembi and Vetr\'ik [On size, order, diameter and edge-connectivity of graphs. Acta Math. Hungar. {\bf 152}, (2017)] that for a connected triangle-free graph of order $n$, diameter $d$ and edge-connectivity $\lambda$, the size is bounded from above by about $\frac{1}{4}\left(n-\frac{(\lambda +c) d}{2}\right)^2+O(n)$, where $c\in\{0, \frac{1}{3}, 1\}$ for different values of $\lambda$. In the second part of this paper we show that this bound by Ali et al. on the size can be improved significantly for a much larger subclass of triangle-free graphs, namely, bipartite graphs of order $n$, diameter $d$ and edge-connectivity $\lambda$. We prove our result only for $\lambda = 2, 3, 4$ because it can be observed from this paper by Ali et al. that for $\lambda\geq 5$, there exists $\ell$-edge-connected bipartite graphs of given order and diameter whose size differs from the maximal size for given minimum degree $\ell$ only by at most a constant. Also, unlike the approach in the proof on the size of triangle-free graphs by Ali et al., our proof employs a completely different technique, which enables us to identify the extremal graphs; hence the bounds presented here are sharp.