For a graph $G=(V,E)$, a subset $M$ of $E$ is called an edge cover of $G$ if $M$ contains at least one incident edge of $v$ for each $v\in V$. Let $cd(G)$ be the maximum number of disjoint edge covers in $G$. It is well known that every simple graph $G$ has $\delta(G)-1\leq cd(G)\leq \delta(G)$, where $\delta(G)$ is the minimum degree of $G$. The problem to determine whether a simple graph $G$ has $cd(G)=\delta(G)$ or not is NP-complete. In this paper, we consider balanced edge cover decompositions. For an integer $k\geq 2$, define that $V_k=\{v\in V: k|d(v)\}$. We show that if $G[V_k]$ is a union of disjoint unicycle graphs or forest, then $G$ has a $k$-edge cover decomposition $E=\cup_{i=1}^{k}E_i$ such that $E_i$ contains at least $\left\lfloor \frac{d(v)}{k}\right\rfloor$ incident edges of $v$ for each $v\in V$ and each $i\in \{1,2,\ldots, k\}$, with only exception that $G[V_k]$ is a union of disjoint cycles and each vertex $v\in V\setminus V_k$ has $d(v)=1\ (mod\ k)$ and $v\in N(V_k)$. This extends the previous results on $G[V_k]$ being a forest or $G$ being peelable.
Joint work with Xia Zhang.