Covering the edges of a graph with perfect matchings

TL;DR

This paper improves Lovász's theorem by showing that for r-graphs, there exists a solution to the edge-covering problem with entries limited to integers or +1/2, linked to a linearly independent set of perfect matchings.

Contribution

It provides a refined solution structure for r-graphs, bounding the number of +1/2 entries and connecting solutions to perfect matchings and graph decomposition.

Findings

Existence of solutions with entries in integers or +1/2

Bound of at most 6k for +1/2 entries, where k is Petersen bricks

Solution corresponds to a linearly independent set of perfect matchings

Abstract

An $r$-graph is an $r$-regular graph with no odd cut of size less than $r$. A well-celebrated result due to Lov\'asz says that for such graphs the linear system $Ax = \textbf{1}$ has a solution in $\mathbb{Z}/2$, where $A$ is the $0,1$ edge to perfect matching incidence matrix. Note that we allow $x$ to have negative entries. In this paper, we present an improved version of Lov\'asz's result, proving that, in fact, there is a solution $x$ with all entries being either integer or $+1/2$ and corresponding to a linearly independent set of perfect matchings. Moreover, the total number of $+1/2$'s is at most $6k$, where $k$ is the number of Petersen bricks in the tight cut decomposition of the graph.