Paths on the Doubly Covered Region of a Covering of the Plane by Unit Discs

Given a covering of the plane by closed unit discs $\mathcal F$ and two points $A$ and $B$ in the region doubly covered by $\mathcal F$, what is the length of the shortest path connecting them that stays within the doubly covered region? This is a problem of G. Fejes-T\'oth and he conjectured that if the distance between $A$ and $B$ is $d$, then the length of this path is at most $\sqrt 2 d+O(1)$. In this paper we give a bound of $2.78 d+O(1)$.