2-Factors in Graphs
Jan van den Heuvel, Bjarne Toft
TL;DR
The paper addresses the classical problem of when a graph contains a $2$-factor, placing the result in a rich historical context of Tutte, Gallai, Belck, and Petersen. It delivers a direct, graph-theoretic proof of the $2$-Factor Theorem via Gallai–Belck alternating chains, and it extends the framework to a complete characterization of maximal graphs without a $2$-factor within the edge-multiplicity bound class $\\mathcal{M}_2$ using a precise structural description and a max-flow formulation. A key consequence is a clean assertion: a $(2k+1)$-regular graph with at most $2k$ leaves has a $2$-factor, generalizing Petersen’s theorem for $3$-regular graphs and connecting to Sylvester’s extremal constructions. The work also highlights Belck’s underappreciated role in factor theory and provides a pathway toward general $k$-factor results, with a clear method for identifying maximal $2$-factor-free graphs and their extremal properties.
Abstract
An account of 2-factors in graphs and their history is presented. We give a direct graph-theoretic proof of the 2-Factor Theorem and a new variant of it, and also a new complete characterisation of the maximal graphs without 2-factors. This is based on the important works of Tibor Gallai on 1-factors and of Hans-Boris Belck on k-factors, both published in 1950 and independently containing the theory of alternating chains. We also present an easy proof that a $(2k+1)$-regular graph with at most $2k$ leaves has a 2-factor, and we describe all connected $(2k+1)$-regular graphs with exactly $2k+1$ leaves without a 2-factor. This generalises Julius Petersen's famous theorem, that any 3-regular graph with at most two leaves has a 1-factor, and it generalises the extremal graphs Sylvester discovered for that theorem.