Connected cubic graphs with the maximum number of perfect matchings

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.

Figures (19)

• Figure 1: The graph $M_5$
• Figure 2: The graph $M_n$ for $n \ge 6$
• Figure 3: Case 1---a connected bipartite cubic graph $G$ with ladder-bridge $xy$
• Figure 4: Subcase 1-1---when $xy$ is next to another ladder-bridge $bd$
• Figure 5: Subcase 1-1---modifying the graph $G$ to $G^\prime$ and $G^{\prime\prime}$

Theorems & Definitions (22)

• Theorem 1.1: Alon--Freidland AF08, 2008
• Remark 1.2
• Theorem 1.3: Galbati Gal81, 1981
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Definition 2.2
• Proposition 2.3