A degree condition implying Ore-type condition for even [2,b]-factors in graphs

For a graph G and even integers b ⩾ a ⩾ 2, a spanning subgraph F of G such that a ⩽ degF (x) ⩽ b and degF (x) is even for all x ∈ 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 max {degG (x),degG (y)}⩾max {2n2+b,3}$\max \{ \deg _G (x),\deg _G (y)\} \ge \max \left\{ {{{2n} \over {2 + b}},3} \right\}$ for any nonadjacent vertices x and y of G. Moreover, we show that for b ⩾ 3a and a > 2, there exists an infinite family of 2-edge-connected graphs G of order n with δ(G) ⩾ a such that G satisfies the condition degG (x)+degG (y)>2ana+b$\deg _G (x) + \deg _G (y) > {{2an} \over {a + b}}$ 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