Gallai's path decomposition in planar graphs

Abstract

In 1968, Gallai conjectured that the edges of any connected graph with $n$ vertices can be partitioned into $\lceil \frac{n}{2}\rceil$ paths. We show that this conjecture is true for every planar graph. More precisely, we show that every connected planar graph except $K_3$ and $K_5^{-}$ ($K_5$ minus one edge) can be decomposed into $\lfloor \frac{n}{2}\rfloor$ paths.