Can we color the edges of any graph $G$ on $2m-1$ vertices with two colors such that any induced subgraph with at least $m$ edges is non-monochromatic?

If true, this would be sharp as shown by the star $K_{1,2m-1}$.

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Can we color the edges of any graph $G$ on $2m-1$ vertices with two colors such that any induced subgraph with at least $m$ edges is non-monochromatic?

If true, this would be sharp as shown by the star $K_{1,2m-1}$.