Let $n\in\mathbb{N}$ be a positive integer. Is there a connected graph $G$ such that $G$ cannot be coloured with less than $n$ colours, and every two maximal [matchings](https://en.wikipedia.org/wiki/Matching_(graph_theory)) have non-empty intersection?


(I take maximality with respect to set inclusion.)



Maximal matchings in connected graphs
Another attempt. Take $K\_{n+1} $ and three paths $abc, ade, afg$, $a\in K\_{n+1} $, $b, c, d, e, f, g\notin K\_{n+1} $. Any maximal matching contains at most one of edges $ab, ad, af$, thus at least two from three edges $bc, de, fg$. Hence any two maximal matchings have a common edge. 


