SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1725-1768, January 2020.
A multigraph $G$ is near-bipartite if $V(G)$ can be partitioned as $I,F$ such that $I$ is an independent set and $F$ induces a forest. We prove that a multigraph $G$ is near-bipartite when $3|W|-2|E(G[W])|\ge -1$ for every $W\subseteq V(G)$, and $G$ contains no $K_4$ and no Moser spindle. We prove that a simple graph $G$ is near-bipartite when $8|W|-5|E(G[W])|\ge -4$ for every $W\subseteq V(G)$, and $G$ contains no subgraph from some finite family $\mathcal{H}$. We also construct infinite families to show that both results are the best possible in a very sharp sense.

中文翻译：

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稀疏图接近二分

SIAM离散数学杂志，第34卷，第3期，第1725-1768页，2020年1月。
如果可以将$ V（G）$划分为$ I，F $，使得$ I $，则多图$ G $几乎是二分的。是一个独立的集合，$ F $会诱导森林。我们证明，当每$ W \ subseteq V（G）$ $ 3 | W | -2 | E（G [W]）| \ ge -1 $且$ G $包含时，多图$ G $是近偶的没有$ K_4 $，也没有Moser主轴。我们证明，对于每个$ W \ subseteq V（G）$和$ G $，当$ 8 | W | -5 | E（G [W]）| \ ge -4 $时，一个简单的图形$ G $几乎是二分图。不包含来自某个有限族$ \ mathcal {H} $的子图。我们还构建了无限的族，以显示两个结果在非常敏锐的意义上都是最好的。