Gallai's path decomposition in planar graphs

TL;DR

This paper proves Gallai's conjecture for all connected planar graphs, showing they can be decomposed into a number of paths close to the conjectured bound, with exceptions for specific small graphs.

Contribution

It establishes the validity of Gallai's path decomposition conjecture for all connected planar graphs, except for two small exceptions.

Findings

Gallai's conjecture holds for all connected planar graphs except K_3 and K_5^-

Every such graph can be decomposed into approximately n/2 paths

The exceptions are specific small graphs K_3 and K_5^-

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.