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

Let ${\mathcal{G}}_{\alpha }$ be a hereditary graph class (i.e, every subgraph of $G_{\alpha }\in {\mathcal{G}}_{\alpha }$ belongs to ${\mathcal{G}}_{\alpha }$) such that every graph $G_{\alpha }$ in ${\mathcal{G}}_{\alpha }$ has minimum degree at most 1, or contains either an edge $uv$ such that $d_{G_{\alpha }}(u)+d_{G_{\alpha }}(v)\le \alpha$ or a 2-alternating cycle. It is proved that every graph in ${\mathcal{G}}_{\alpha }$ ($\alpha \ge 5$) with maximum degree $\mathrm{\Delta }$ 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$\Delta(H)\leq \alpha-5$.