On Gallai's conjecture for series-parallel graphs and planar 3-trees

TL;DR

This paper proves Gallai's conjecture for series-parallel graphs and improves the upper bound for planar 3-trees, demonstrating more efficient path covers for these graph classes.

Contribution

It establishes Gallai's conjecture for series-parallel graphs and provides a better bound for planar 3-trees, advancing understanding of path covers in these graphs.

Findings

Gallai's conjecture holds for series-parallel graphs.

A path cover with at most  n/2  paths exists for series-parallel graphs.

Planar 3-trees have a path cover with at most  5n/8  paths, improving previous bounds.

Abstract

A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected $n$-vertex graph has a path cover with at most $\lceil n/2 \rceil$ paths. We prove Gallai's conjecture for series-parallel graphs. For the class of planar 3-trees we show how to construct a path cover with at most $\lfloor 5n/8 \rfloor$ paths, which is an improvement over the best previously known bound of $\lfloor 2n/3 \rfloor$.