Solitude patterns of $r$-graphs via CLM's dependence relation

Abstract

Recently, Goedgebeur, Mazzuoccolo, Renders, Wolf and the second author [Cubic graphs with edges in exactly one perfect matching, J. Graph Theory 112 (2026), 276-289] proved that every $3$-connected $3$-regular graph has at most six solitary edges; an edge is solitary if it participates in precisely one perfect matching. They also gave complete characterizations of $3$-connected $3$-regular graphs that have $k$ solitary edges for each $k \ge 3$. A connected $r$-regular graph, where $r \geq 3$, is an $r$-graph if each odd cut has at least $r$ edges; for instance, $3$-graphs are precisely the $2$-connected $3$-regular graphs. We generalize the bound and characterizations, mentioned in the preceding paragraph, to $3$-edge-connected $r$-graphs of order four or more; in particular, for $r \geq 4$, we establish stronger bounds:\ (i) at most four solitary edges, and (ii) if the graph is simple then it is devoid of solitary edges. Apart from characterizing all $3$-edge-connected $r$-graphs that have three or more solitary edges, for the case $r=3$, we provide a recursive characterization of those that have precisely two solitary edges such that they lie in the same perfect matching. Finally, for all $r$-graphs (not necessarily $3$-edge-connected), we establish that: (i) every such graph, except for a few small graphs, has at most $\frac{n}{2}$ solitary edges, and (ii) every such graph decomposes uniquely into $3$-edge-connected $r$-graphs, and its solitary edges may be computed recursively via this decomposition. Our proofs and insights rely heavily on the dependence and mutual dependence relationships introduced and investigated by Carvalho, Lucchesi and Murty [Ear decompositions of matching covered graphs, Combinatorica 19 (1999), 151-174].