Speaker: Nishad Kothari
Date: Sep 10, 2025
Time: 03:30 PM
Venue: CS25
Abstract:
The study of perfect matchings (aka 1-factors) goes back all the way to the Four Color Conjecture (1852) and related developments in the late 1800s due to Tait (1880) and Petersen (1891). The former established that the Four Color Conjecture is equivalent to the statement that every 2-connected 3-regular planar graph is 3-edge-colorable (or, in other words, has three disjoint perfect matchings), whereas the latter proved that every 2-connected 3-regular graph has a perfect matching (that is, a spanning 1-regular subgraph).
One of the earliest breakthroughs was Tutte’s 1-factor Theorem (1947) that gave a complete characterization of matchable graphs — that is, graphs that have a perfect matching.
Using Tutte’s Theorem, one may easily deduce a strengthening of the aforementioned result of Petersen: in a 2-connected 3-regular graph, each edge belongs to some perfect matching. This leads to one natural direction of research that is discussed below. For a graph G, an edge e is solitary if it belongs to precisely one perfect matching. For example, each edge of K4 is solitary, whereas no edge of the Petersen graph is solitary.
Recently, Goedgebeur, Mattiolo, Mazzuoccolo, Renders and Wolf (http://arxiv.org/abs/2402.08538, 2024) proved that every 3-connected 3-regular graph has at most six solitary edges, and they gave a complete characterization of all such graphs that have three or more solitary edges.
In a joint work with a recently graduated MS scholar D. V. V. Narayana, Davide Mattiolo, and a former intern Kalyani Gohokar, we (http://arxiv.org/abs/2409.00534, 2025) generalize their results to 3-edge-connected r-graphs; when r=3, this is precisely the class of 3-connected 3-regular graphs. Our main tool is the dependence relation introduced by Carvalho, Lucchesi and Murty (Combinatorica, 1999).
In this talk, I will discuss all of the above in as much detail as possible within the timeframe of one hour. No prior background, except for basic graph theory knowledge, is required.