A Degree Condition Implying Ore-Type Condition for Even [2, b]-Factors in Graphs

For a graph $G$ and even integers $ b \ge a \ge 2 $, a spanning subgraph $F$ of $G$ such that $ a \le \text{deg}_F (x) \le b $ and $ \text{deg}_F (x) $ is even for all $ x \in V (F) $ is called an even $[a, b]$-factor of $G$. In this paper, we show that a 2-edge-connected graph $G$ of order $n$ has an even $[2, b]$-factor if $ \text{max} \{ \text{deg}_G (x) , \text{deg}_G (y) \} \ge \text{max} \{ \frac{2n}{2+b} , 3 \} $ for any nonadjacent vertices $x$ and $y$ of $G$. Moreover, we show that for $ b \ge 3a$ and $a > 2$, there exists an infinite family of 2-edge-connected graphs $G$ of order $n$ with $ \delta (G) \ge a$ such that $G$ satisfies the condition $ \text{deg}_G (x) + \text{deg}_G (y) > \frac{2an}{a+b} $ for any nonadjacent vertices $x$ and $y$ of $G$, but has no even $[a, b]$-factors. In particular, the infinite family of graphs gives a counterexample to the conjecture of Matsuda on the existence of an even $[a, b]$-factor.