Dirac's theroem for bipartite graphs

Question

The theorem of Dirac that any graph $G$ on $n\geq 3$ vertices with minimum degree $\delta(G)≥n/2$ contains a Hamilton cycle is one of the classical results of graph theory.

Is there are analogous version which correlates the minimum degree of the vertices in a bipartite graph to guarantee a Hamiltonian cycle ( or alternatively a perfect matching)?

Answer 1

Yes, there is.

We have the following theorem of Bondy (1969):

Let $G = (X,Y,E)$ be a simple bipartite graph with $|X|=|Y|=n\ge 2$. Let the vertices of $X$ and $Y$ be indexed with increasing degrees (that is, $X = \{x_1, \dots, x_n\}$ with $\deg(x_1) \le \deg(x_2) \le \dots \le \deg(x_n)$ and $Y = \{y_1, \dots, y_n\}$ with $\deg(y_1) \le \deg(y_2) \le \dots \le \deg(y_n)$). If $$\deg(x_j) \le j, \deg(y_k) \le k \implies \deg(x_j) + \deg(y_k) \ge n+1$$ then $G$ has a Hamiltonian cycle.

This is much more complicated than Dirac's theorem, but as a weaker corollary, we have:

Let $G = (X,Y,E)$ be a simple bipartite graph with $|X|=|Y|=n\ge 2$. If $\delta(G) \ge \frac{n+1}{2}$, then $G$ has a Hamiltonian cycle.

Just minimum degree $\frac n2$ would not be enough, since then $G$ could be two disjoint copies of $K_{n/2, n/2}$.

I'm getting Bondy's theorem from Chapter 10 of Berge's Graphs and Hypergraphs, which is my go-to for Hamiltonian cycle condition theorems.