2 problems

Let $G$ be a finite simple triangle-free planar graph. A partition of $G$ into an independent set and a forest means that there is a partition $(V_1,V_2)$ of $V(G)$ such that…

Let $k$ and $d$ be positive integers. For a graph $G$, call it $(k,d)$-sparse if every subgraph $H$ satisfies … Call a subgraph $H$ $(k+1)$-overfull if $e(H)>(k+1)(v(H)-1)$. Sparse…