3 problems

For a graph $G$ on $n$ vertices, write $e(G)$ for its number of edges and set … For nonempty vertex subsets $X,Y\subseteq V(G)$, let … Let $\sigma_{2}(G)$ denote the second-largest…

Cycle-factor extension conjecture. If

Ai–Guo–Freschi–Lo conjecture. Let $G$ be an oriented graph on $n\geq 3$ vertices. If $\sigma_2(G)\geq n$, then $G$ contains a Hamilton cycle $C$ such that