Bricks and conjectures of Berge, Fulkerson and Seymour

TL;DR

This paper investigates properties of r-graphs related to edge-coloring and perfect matchings, proving that minimal counterexamples are bricks and disproving a Fan-Raspaud conjecture variant.

Contribution

It demonstrates that the smallest counterexamples to Seymour's conjectures are bricks and provides a disproof of a Fan-Raspaud conjecture variant.

Findings

Minimum counter-examples are bricks.

Disproved a variant of Fan-Raspaud conjecture.

Confirmed structural properties of r-graphs related to Seymour's conjectures.

Abstract

An $r$-graph is an $r$-regular graph where every odd set of vertices is connected by at least $r$ edges to the rest of the graph. Seymour conjectured that any $r$-graph is $r+1$-edge-colorable, and also that any $r$-graph contains $2r$ perfect matchings such that each edge belongs to two of them. We show that the minimum counter-example to either of these conjectures is a brick. Furthermore we disprove a variant of a conjecture of Fan, Raspaud.