| 李宁,侯新民.由圈长分布确定的二部图~$K_{n,n r}-A\ (|A|\leq 3)$[J].数学研究及应用,2009,29(4):571~579 |
| 由圈长分布确定的二部图~$K_{n,n r}-A\ (|A|\leq 3)$ |
| Bipartite Graphs $K_{n,n r}-A~(|A|\leq 3)$ Determined by Their Cycle Length Distributions |
| 投稿时间：2007-06-27 修订日期：2008-01-02 |
| DOI：10.3770/j.issn:1000-341X.2009.04.001 |
| 中文关键词: 圈长分布 二部分图. |
| 英文关键词:cycle length distribution bipartite graphs. |
| 基金项目:国家自然科学基金(Nos.10701068;10671191). |
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| 中文摘要: |
| 阶为~$n$~的图~$G$~的圈长分布为序列~$(c_1(G), \ldots, c_n(G))$,其中~$c_i(G)$~是图$G$中长为~$i$~的圈的数目.一般来说,满足圈长分布~$(c_1(G), \dots,c_n(G))$~的图~$G$~不唯一.如果满足圈长分布~$(c_1(G), \ldots,c_n(G))$~的图~$G$~是唯一的,就称图~$G$~是由它的圈长分布确定的.设~$K_{n,n r}$~为完全二部分图,~$A\subseteq E(K_{n,n r})$.本文得到: 设~$s>1$~为整数.(1) 当~$r=2s, n>s(s-1) 2|A|$~时,$K_{n,n r}-A\ (A\subseteq E(K_{n,n r}),|A|\leq 3)$~是由它的圈长分布确定的; (2) 当~$r=2s 1,n>s^2 2|A|$~时, $K_{n,n r}-A\ (A\subseteq E(K_{n,n r}),|A|\leq 3)$~是由它的圈长分布确定的. |
| 英文摘要: |
| The cycle length distribution of a graph $G$ of order $n$ is a sequence $(c_1(G), \dots, c_n(G))$, where $c_i(G)$ is the number of cycles of length $i$ in $G$. In general, the graphs with cycle length distribution $(c_1(G), \dots, c_n(G))$ are not unique. A graph $G$ is determined by its cycle length distribution if the graph with cycle length distribution $(c_1(G), \dots, c_n(G))$ is unique. Let $K_{n,n r}$ be a complete bipartite graph and $A\subseteq E(K_{n,n r})$. In this paper, we obtain: Let $s>1$ be an integer. (1) If $r=2s, n>s(s-1) 2|A|$, then $K_{n,n r}-A\ (A\subseteq E(K_{n,n r}),|A|\leq 3)$ is determined by its cycle length distribution; (2) If $r=2s 1, n>s^2 2|A|$, $K_{n,n r}-A\ (A\subseteq E(K_{n,n r}),|A|\leq 3)$ is determined by its cycle length distribution. |
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