**Question**For every does there exists a such that every graph with average degree smaller than and girth at least has a matching-cut?

Let be a graph. A matching is a *matching-cut* if there exists a set such that . Graphs having a matching-cut are called *decomposable.*

It is known that every graph with is decomposable [BFP11].

## Bibliography

[C84] V. Chvátal, Recognizing decomposable graphs, J Graph Theory 8 (1984), 51–53

[BFP11] P. Bonsma, A. Farley, A. Proskurowski, Extremal graphs having no matching cuts, J Graph Theory (2011)

* indicates original appearance(s) of problem.