Graphs in which all maximal matchings intersect Let $G=(V,E)$ be a simple, undirected graph. A matching is a set $M\subseteq E$ consisting of pairwise disjoint edges. We say $M$ is maximal if it is maximal amongst all matchings in $G$ with respect to $\subseteq$.
Given $n\in\mathbb{N}$ is there a graph $G$ such that $\chi(G)\geq n$ and every two maximal matchings have non-empty intersection?
 A: Affirmative: take $G$ to be the disjoint union of $K_n$ and $K_2$, then the $K_n$ forces $\chi(G)\geq n$, while any maximal matching contains the $K_2$ edge.
OK, that was cheating. I suppose you want a connected example.
I don't know the answer there, but can give a negative answer if you replace the condition $\chi(G)\geq n$ with $|G|\geq n$ and minimum degree of a vertex is at least $|G|/2$. For in this case it is known that $G$ has at least $\lfloor 5n/224\rfloor$ edge-disjoint Hamiltonian cycles (Nash-Williams). So if $n$ is even and at least $90$, then $G$ will have at least $2$ edge-disjoint Hamiltonian cycles. If you take every other edge in an even-length Hamiltonian cycle you will get a maximal matching. Since you have at least $2$ edge-disjoint Hamiltonian cycles, you end up with two edge-disjoint maximal matchings.
Edit. I guess you only need one even-length Hamiltonian cycle to get $2$ edge-disjoint maximal matchings: for one take every other edge in the Hamiltonian cycle and for the other take edges of the cycle not used in the first matching.
A: Finding such a graph that is connected: Consider the following graph $G$ consisting of:


*

*A clique $K_n$ of $n$ vertices $v_1,\ldots v_n$.

*For some integer $D \geq 4$, $D$ paths $P_{di}$; $i=1,\ldots, n$ and $d=1,\ldots D$ where $P_{di}$ is a path $v_iv_{di1}v_{di2}$ [the $v_{di1}$s and the $v_{di2}$s are all distinct from each other and are NOT in the clique $K_n$ as in 1].
Then one can check that for any two maximal matchings $M_1$ and $M_2$ there is a $d$ and an $i$ s.t. the edge $v_{di1}v_{di2}$ is in both $M_1$ and $M_2$. [So any two maximal matchings intersect. The stronger result of there being a common edge to all maximal matchings does not hold here though.]
