# Properties of bipartite graphs

For a connected bipartite graph $G$ are the two following properties equivalent:

1)Every minimal cycle in $G$ has length 4, that is every cycle of length strictly greater than 4 can be divided in cycles of length 4 ($Z^d$ is such a graph).

2) For all triplet of vertices $\{ v_1, v_2, v_3 \}$ in $G$ there exist three geodesics from $v_1$ to $v_2$, $v_1$ to $v_3$ and $v_2$ to $v_3$ respectively, whose union form a subtree of $G$.

If yes, is there any reference for this? If no what would be a counter example?

• I am sincerely sorry for the approximative formulation, I am not very knowledgeable in graph theory. What I am looking for is for a different characterization of the property 2 (I need to check if some particular Cayley graph has it). Clearly chordal bipartite or having minimal cycles of length 4 does not work since the discretization of $S^2$ into a grid does not have property 2. I was hoping to find a reference in the literature. Commented Feb 6, 2016 at 22:33