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Let $P,Q$ be longest paths in a simple graph. We analyze the possibleconnections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ andintroduce the notion of a bi-traceable graph. We use the results for all thepossible configurations of the intersection points when $\#V(P)\cap V(Q)\le 5$in order to prove that if the intersection of three longest paths $P,Q,R$ isempty, then $\#(V(P)\cap V(Q))\ge 6$. We also prove Hippchen's conjecture for$k\le 6$: If a graph $G$ is $k$-connected for $k\le 6$, and $P$ and $Q$ arelongest paths in $G$, then $\#(V(P)\cap V(Q))\ge 6$.

Bi-traceable graphs, the intersection of three longest paths and Hippchen's conjecture