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Let $\mathcal{G}_{\alpha}$ be a hereditary graph class (i.e, every subgraphof $G_{\alpha}\in \mathcal{G}_{\alpha}$ belongs to $\mathcal{G}_{\alpha}$) suchthat every graph $G_{\alpha}$ in $\mathcal{G}_{\alpha}$ has minimum degree atmost 1, or contains either an edge $uv$ such that$d_{G_{\alpha}}(u)+d_{G_{\alpha}}(v)\leq \alpha$ or a 2-alternating cycle. Itis proved that every graph in $\mathcal{G}_{\alpha}$ ($\alpha\geq 5$) withmaximum degree $\Delta$ can be edge-partitioned into two forests $F_1$, $F_2$and a subgraph $H$ such that $\Delta(F_i)\leq\max\{2,\lceil\frac{\Delta-\alpha+6}{2}\rceil\}$ for $i=1,2$ and $\Delta(H)\leq\alpha-5$.

A note on the edge partition of graphs containing either a light edge or an alternating 2-cycle