Two-Edge-Connected $[2,k]$-Factors in Graphs.

Abstract

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 $\delta (F)\ge a$ and maximum degree $\mathrm{\Delta }(F)\le b$.
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 $k\ge 2$ be an integer. If ${\sigma }_2(G)\ge 4n/(k+2)$, 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 $\delta (G)\ge 2n/(k+2)$, then $G$ has a 2-edge-connected $[2,k]$-factor.