# How does symmetric difference of b-matchings look like?

It is easy to see that symmetric difference of two matchings are cycles or simple paths. But what about b-matchings? Is there anything known about how they look? Even for restricted cases such as bipartite graphs $$G=(U,V,E)$$ with $$b(v)=1\ \ \forall v\in U$$

$$def: b-matching$$: Given a graph $$G=(V,E)$$ and an upper bound $$b:V\to \mathbb{Z}^+$$ for each vertex, a $$b-matching$$ is a sub-graph $$H$$ of $$G$$ such that $$deg_H(v) \le b(v) \ \ \forall v\in V$$. (Regular matching is a special case when $$b(v)=1$$)

• My intuition is that the union (and likely also the symmetric difference) of three or more matchings can be very complicated. For example, Conjecture 2 in this paper is that every cubic, bridgeless graph is the union of 5 perfect matchings. Jan 5 at 19:26