通过CLM的依赖关系研究r-图的孤立边模式
Solitude patterns of $r$-graphs via CLM's dependence relation
AI总结：
该研究将3-连通3-正则图的孤立边界与刻画推广到3-边连通r-图，对不同r值建立了更严格的孤立边界，还给出了r=3时特定孤立边情况的递归刻画，证明了所有r-图的孤立边相关性质。
摘要：
近期，Goedgebeur、Mazzuoccolo、Renders、Wolf与第二作者在论文《Cubic graphs with edges in exactly one perfect matching》（发表于《J. Graph Theory》第112卷，2026年，第276-289页）中证明：每个3-连通3-正则图至多有6条孤立边；一条边若仅参与一个完美匹配，则称为孤立边。他们还对每个k≥3时，具有k条孤立边的3-连通3-正则图给出了完整刻画。一个连通r-正则图（r≥3）若满足每个奇割至少有r条边，则称为r-图；例如，3-图恰好是2-连通3-正则图。我们将上述提到的界和刻画推广到阶数不小于4的3-边连通r-图；特别地，对于r≥4，我们建立了更强的界：(i) 至多有4条孤立边；(ii) 若该图是简单图，则它没有孤立边。除了对所有具有3条或更多孤立边的3-边连通r-图进行刻画外，对于r=3的情况，我们还对恰好有两条孤立边且这两条边位于同一个完美匹配中的r-图给出了递归刻画。最后，对于所有r-图（不一定是3-边连通的），我们证明：(i) 除了少数几个小型图外，每个此类图至多有n/2条孤立边；(ii) 每个此类图可唯一分解为3-边连通r-图，且其孤立边可通过该分解递归计算。我们的证明和见解大量依赖于Carvalho、Lucchesi和Murty在论文《Ear decompositions of matching covered graphs》（发表于《Combinatorica》第19卷，1999年，第151-174页）中引入并研究的依赖关系和相互依赖关系。
英文摘要：
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].
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