The logical strength of K\"onig's edge coloring theorem

TL;DR

This paper investigates the computability and logical strength of K"onig's edge coloring theorem, establishing its equivalence to certain subsystems of reverse mathematics and exploring related open problems.

Contribution

It analyzes the theorem within computability theory and reverse mathematics, providing new proofs and connections to logical systems like WKLo.

Findings

Computable bipartite graphs with degree ≤ n have computable (n+1)-edge colorings.

The existence of an n-color edge coloring is equivalent to WKLo over RCAo.

Links to Vizing's theorem and Birkhoff's theorem are discussed.

Abstract

K\"onig's edge coloring theorem says that a bipartite graph with maximal degree $n$ has an edge coloring with no more than $n$ colors. We explore the computability theory and Reverse Mathematics aspects of this theorem. Computable bipartite graphs with degree bounded by $n$ have computable edge colorings with $n+1$ colors, but the theorem that there is an edge coloring with $n$ colors is equivalent to WKLo over RCAo. This gives an additional proof of a theorem of Hirst: WKLo is equivalent over RCAo to the principle that every countable bipartite $n$-regular graph is the union of $n$ complete matchings. We describe open questions related to Vizing's edge coloring theorem and a countable form of Birkhoff's theorem.