In this paper, we give the following result: If $D$ is a digraph of order $n$, and if $d_{D}^{+}(u) + d_{D}^{-}(v) \ge n$ for every two distinct vertices $u$ and $v$ with $(u, v) \notin A(D)$, then $D$ has a directed 2-factor with exactly $k$ directed cycles of length at least 3, where $n \ge 12k+3$. This result is equivalent to the following result: If $G$ is a balanced bipartite graph of order 2n with partite sets $X$ and $Y$, and if $d_{G}(x)+d_{G}(y) \ge n + 2$ for every two vertices $x \in X$ and $y \in Y$ with $xy \notin E(G)$, then for every perfect matching $M$, $G$ has a 2-factor with exactly $k$ cycles of length at least 6 containing every edge of $M$, where $n \ge 12k+3$. These results are generalizations of theorems concerning Hamilton cycles due to Woodall [ Proc. Lond. Math. Soc., 24 (1972), pp. 739--755] and Las Vergnas [ Problémes de couplages et problémes hamiltoniens en théorie des graphes, Ph.D. thesis, University of Paris, 1972], respectively.

It is known that a 3-edge-connected graph has a spanning even subgraph in which every component contains at least five vertices, and the lower bound is best possible. A natural question arises of whether we can improve the lower bound by changing the spanning property with the dominating property. In this paper, we show that a 3-edge-connected cubic graph has a dominating even subgraph in which every component contains at least six vertices.

In 1986, Thomassen posed the following conjecture: every 4-connected line graph has a Hamiltonian cycle. As a possible approach to the conjecture, many researchers have considered statements that are equivalent or related to it. One of them is the conjecture by Bondy: there exists a constant $c_0$ with $0 < c_0 \leq 1$ such that every cyclically 4-edge-connected cubic graph $H$ has a cycle of length at least $c_0 |V(H)|$. It is known that Thomassen's conjecture implies Bondy's conjecture, but nothing about the converse has been shown. In this paper, we show that Bondy's conjecture implies a slightly weaker version of Thomassen's conjecture: every 4-connected line graph with minimum degree at least 5 has a Hamiltonian cycle.

For a subset $A$ of the set of positive integers, a graph $G$ is called $A$-coverable if $G$ has a cycle (a subgraph in which all vertices have even degree) which intersects all edge-cuts $T$ in $G$ with $|T| \in A$, and $A$ is said to be coverable if all graphs are $A$-coverable. As a possible approach to the dominating cycle conjecture, Kaiser and Škrekovski conjectured in [SIAM J. Discrete Math., 22 (2008), pp. 861--874] that $\mathbb{N} +3$ is coverable, where $\mathbb{N} +3 = \{4,5,6, \ldots\}$. In this paper, we disprove Kaiser and Škrekovski's conjecture by showing that there exist infinitely many graphs which are not $\{4,5\}$-coverable.