The connectivity of a bipartite graph and its bipartite complementary graph

TL;DR

This paper establishes optimal Nordhaus-Gaddum type inequalities for the connectivity of bipartite graphs and their bipartite complements, expanding understanding of graph invariants in bipartite structures.

Contribution

It introduces the first bounds for the connectivity of bipartite graphs and their bipartite complements, proving these bounds are tight.

Findings

Derived the best possible inequalities for bipartite graph connectivity.

Extended Nordhaus-Gaddum inequalities to bipartite and bipartite complementary graphs.

Proved the inequalities are sharp and cannot be improved.

Abstract

In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph $G$ and the same invariant in the complement $G^c$ of $G$ is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph $G=G[X,Y]$ with bipartition ($X,Y$), its bipartite complementary graph $G^{bc}$ is a bipartite graph with $V(G^{bc})=V(G)$ and $E(G^{bc})=\{xy:\ x\in X,\ y\in Y$ and $xy \notin E(G)\}$. In this paper, we obtain the Nordhaus-Gaddum type inequalities for connectivity of bipartite graphs and its bipartite complementary graphs. Furthermore, we prove that these inequalities are best possible.