Hamiltonian

\[\sigma_{1,1}(G) := \min \{ d_{G}(x) + d_{G}(y) \mid x \in X, y \in Y, (x,y) \notin E(G) \} .\]

Let

\[\Delta(\]

Theorem

• $X, Y$,
  • vertexes
  • $X \cap Y \neq \emptyset$
• $G[X, Y]$,
  • bipartite graph
• $|X| = |Y| = n \ge 3$,
• $\sigma_{1, 1}(G) \ge n + 1$,

For all $u \in V(G)$ and $k \in \mathbb{N}$ $3 \le k \le n$, there is a cycle of $u$ whose length $2k$.

proof

$\Box$

Theorem

proof

$\Box$

Reference