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arXiv 0906.3873math.COmath-phmath.MP
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最大度小的线图的完美匹配

Perfect matchings of line graphs with small maximum degree

Weigen Yan, Fuji Zhang

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20 pages, 12 figures
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摘要：

设$G$是一个连通图，其顶点集为$V(G)=\{v_1,v_2,...,v_ν\}$，可以有多条边但没有环，其中$i=1,2,...,ν$为$2\leq d_G(v_i)\leq 3$，其中$d_G(v)$表示$G$的顶点$v$的度.证明了当$G$的边数为偶数时，$G$的线图的完美匹配数等于$2^{n/2+1}$，其中$n$是$G$的3度顶点数.作为推论，我们证明了具有$n$顶点的连通三次线图的完美匹配数等于$2^{n/6+1}$，如果$n>4$，这意味着Lovász和Plummer的猜想对连通三次线图成立.作为应用，我们在统计物理的背景下列举了Kagomé格、$3.12.12$格和Sierpinski垫片的二维完美匹配.

英文摘要：

Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,...,v_ν\}$, which may have multiple edges but have no loops, and $2\leq d_G(v_i)\leq 3$ for $i=1,2,...,ν$, where $d_G(v)$ denotes the degree of vertex $v$ of $G$. We show that if $G$ has an even number of edges, then the number of perfect matchings of the line graph of $G$ equals $2^{n/2+1}$, where $n$ is the number of 3-degree vertices of $G$. As a corollary, we prove that the number of perfect matchings of a connected cubic line graph with $n$ vertices equals $2^{n/6+1}$ if $n>4$, which implies the conjecture by Lovász and Plummer holds for the connected cubic line graphs. As applications, we enumerate perfect matchings of the Kagomé lattices, $3.12.12$ lattices, and Sierpinski gasket with dimension two in the context of statistical physics.

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