It is well-known that some of the most difficult open problems in structural graph theory --- for instance, the Cycle Double Cover (CDC) Conjecture and Tutte's 5-flow (T5F) Conjecture --- reduce to \(2\)-connected \(3\)-regular graphs. Additionally, there are many notorious open problems pertaining to this class of graphs --- for instance, the Berge-Fulkerson (BF) Conjecture --- that have resisted all attempts thus far. For some such conjectures (including BF), researchers have proposed generalizations to \(r\)-graphs --- that is, to those \(r\)-regular graphs (where \(r \geq 3\)) in which each odd cut has cardinality \(r\) or more.
In fact, it is worth noting that for most of these open problems (including the aforementioned CDC, T5F and BF), the real difficulty lies in dealing with those graphs that are not \(r\)-edge-colorable (aka \emph{Class-2} graphs, or \emph{snarks} in the case of \(r=3\)).
One may easily prove using Tutte's \(1\)-factor Theorem that every \(r\)-graph is matching covered --- that is, it is connected, and each edge lies in some perfect matching. Although this fact is well-known, there have been very few attempts at applying the extensive theory of matching covered graphs (see \emph{Perfect Matchings: A Theory of Matching Covered Graphs} by Lucchesi and Murty, \url{https://link.springer.com/book/10.1007/978-3-031-47504-7}, Springer 2024) to \(r\)-graphs (except perhaps in the case of \(r=3\)).
In this talk, I shall discuss three recent works that apply the theory of matching covered graphs to (some/all) \(r\)-graphs. One of them is a joint work with Narayana, Mattiolo and Gohokar (see \url{https://arxiv.org/abs/2409.00534}), another is a joint work with Joshi, Raghul and Diwan (see \url{https://arxiv.org/abs/2407.05264}) and the third one is an-going work with Narula; the latter two focus on \(r=3\) (including snarks), whereas the first one applies to all \(r\)-graphs and generalizes most of the results of Goedgebeur, Mattiolo, Mazzuoccolo, Renders and Wolf (see \url{https://arxiv.org/abs/2402.08538}).
