The Merino-Welsh Conjecture holds for Series-Parallel Graphs

TL;DR

This paper proves the Merino-Welsh conjecture for series-parallel graphs, establishing a key inequality relating spanning trees, cyclic, and acyclic orientations within this class.

Contribution

The paper provides the first proof of the Merino-Welsh conjecture specifically for series-parallel graphs, expanding the class of graphs where the conjecture is verified.

Findings

The conjecture holds for all series-parallel graphs.

The proof confirms the inequality relating spanning trees and orientations.

This advances understanding of graph orientation properties.

Abstract

The Merino-Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series-parallel graphs.