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Let \(\sigma_2(G) = \min\{d_G(u)+d_G(v) | u,v \in V(G), u,v \notin E(G)\}\) for a non-complete graph \(G\). An \([a, b]\)-factor of \(G\) is a spanning subgraph \(F\) with minimum degree \(\delta(F) \geq a\) and maximum degree \(\Delta(F) \leq b\).
In this note, we give a partially positive answer to a conjecture of M. Kano. We prove the following results:
Let \(G\) be a 2-edge-connected graph of order \(n\) and let \(k \geq 2\) be an integer. If \(\sigma_2(G) \geq {4n}/{(k +2)}\), then \(G\) has a 2-edge-connected \([2, k]\)-factor if \(k\) is even and a 2-edge-connected \([2, k + 1]\)-factor if \(k\) is odd.
Indeed, if \(k\) is odd, there exists a graph \(G\) which satisfies the same hypotheses and has no 2-edge-connected \([2, k]\)-factor.
Nevertheless, we have shown that if \(G\) is 2-connected with minimum degree \(\delta(G) \geq {2n}/{(k +2)}\), then \(G\) has a 2-edge-connected \([2, k]\)-factor.