CLM's dependence relation, solitary patterns and $r$-graphs

Abstract: A connected r-regular graph, where $r \geq 3$, is an r-graph if each odd cut has at least r edges. Every r-graph is matching covered - a connected graph whose each edge participates in some perfect matching. We set out to: (i) characterize solitary edges - those edges that participate in only one perfect matching, and (ii) upper bound the number of such edges. Two edges are mutually dependent if every perfect matching containing either of them also contains the other. Clearly, this is an equivalence relation and induces a partition of E(G). It is worth noting that if any member of an equivalence class is solitary then so is every member; we refer to such an equivalence class as a solitary class. This immediately brings us to the notion of solitary pattern of a matching covered graph - the sequence of cardinalities of its solitary classes in nonincreasing order. Clearly, n/2 is an upper bound on the cardinality of any equivalence class, and if equality holds then each largest equivalence class is a solitary class. We provide a characterization of all matching covered graphs that attain this upper bound. However, all such graphs, of order six or more, contain 2-cuts. On the other hand, using a result of Lucchesi and Murty, we deduce that in a 3-edge-connected r-graph, every solitary class has cardinality one or two. We prove that the distance between any two solitary classes in any 3-edge-connected r-graph is at most three; furthermore, if the order is four or more, we establish that the number of solitary classes is at most three and equality holds if and only if r = 3. Ergo, every 3-edge-connected r-graph, of order four or more, has one of the following ten solitary patterns: (2, 2, 2), (2, 2, 1), (2, 1, 1), (1, 1, 1), (2, 2), (2, 1), (2), (1, 1), (1) or (). We provide complete characterizations of 3-edge-connected r-graphs that have one of the first six solitary patterns.