由圈长分布确定的二部图~$K_{n,n r}-A\ (|A|\leq 3)$

The cycle length distribution of a graph G of order n is a sequence （c1 （G）,…, cn （G））, where ci （G） is the number of cycles of length i in G. In general, the graphs with cycle length distribution （c1（G） ,…,cn（G）） are not unique. A graph G is determined by its cycle length distribution if the graph with cycle length distribution （c1 （G）,…, cn （G）） is unique. Let Kn,n＋r be a complete bipartite graph and A lohtaib in E（Kn,n＋r）. In this paper, we obtain： Let s 〉 1 be an integer. （1） If r = 2s, n 〉 s（s - 1） ＋ 2｜A｜, then Kn,n＋r - A （A lohtain in E（Kn,n＋r）,｜A｜ ≤ 3） is determined by its cycle length distribution; （2） If r = 2s ＋ 1,n 〉 s^2 ＋ 2｜A｜, Kn,n＋r - A （A lohtain in E（Kn,n＋r）, ｜A｜ ≤3） is determined by its cycle length distribution.

Bipartite Graphs $K_{n,n r}-A~(|A|\leq 3)$ Determined by Their Cycle Length Distributions

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.