Abstract:It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well. In addition, it is shown that the number of perfect matchings in any cubic graph $G$ equals the expected value of a random variable defined on all $2$-colorings of edges of $G$. Finally, an improved lower bound on the maximum number of cycles in a cubic graph is provided.
From: Dongryul Kim [view email]
[v1]
Wed, 24 Jun 2020 04:04:39 UTC (18 KB)
[v2]
Fri, 7 Aug 2026 08:08:58 UTC (19 KB)
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