# Sparse graphs are near-bipartite

## TL;DR

This paper characterizes near-bipartite multigraphs and simple graphs using local density conditions and forbidden subgraphs, providing sharp bounds and constructions to demonstrate optimality.

## Contribution

It establishes new necessary and sufficient conditions for near-bipartiteness based on local inequalities and forbidden subgraphs, with optimal bounds and explicit constructions.

## Key findings

- Characterization of near-bipartite multigraphs with local density condition
- Characterization of near-bipartite simple graphs with forbidden subgraphs
- Construction of infinite families showing bounds are sharp

## Abstract

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 best possible in a very sharp sense.