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z`SKNKy_>3#IrLQ1GWk;N=RED*^AgsWMYG?}jTpjU4xrJ8lsk%AlMRE+!6EW>zMne`9tNQxj7|R%X8cb460^D**@x zd(!MHIe_W^OSX|ubIS!PB8Bc@gDE2U2?*Bg!-Ijfrwil}ZQy!o1mQa(KiUjUEMUDZ3Im~dnv7nG+@^0%vh!XeTUsH3{+=ZT+M^8({TtE)|mskB(P0UT*u3bLD%Ul4% gk\Delta$, 所以$\chi'=\Delta+1$. \item[3.] 如果所有cycle长度都是偶数, $G$是二分图, $\chi'=\Delta$. 如果所有cycle长度都是奇数, 由于两个cycle的对称差还是cycle, 观察到任意两个奇环不能共用边,只能共用点. 将每个奇环看成一个点, 共用顶点的奇环之间连边, 得到的图一定是 forest, 否则包含简单偶环. 由于每个奇环只需要3种颜色即可染色且奇环之间形成树(要求$G$联通), 容易看出$\chi'\leq \Delta$. \item[4.] 对$n$归纳, base case是$P_3$,显然成立. 假设对所有$k\leq n-1$阶满足条件的简单图都有$\chi'=k-1$. 现在考虑$k$阶满足条件的简单图$G$, 其中$\deg(v)=k-1$. $G-v$满足归纳假设, 有一个$k-2$边染色, 所以一定存在一个$k-1$边染色. 重复使用$k-1$次引理1容易推出$G$也可$k-1$边染色. -\item[5.] +\item[5.] 如果$G$不含任何奇圈, 那么是二分图, $\chi=2$. 如果$G$只有一个奇圈, 显然$\chi=3$. 如果$G$ 有多个奇圈, 取其中一个,记为$C$, $C$可以3染色. 考虑$G\setminus C$, 由于任何两个奇圈都共用顶点, 所以$G\setminus C$没有奇圈, 是二分图, 可以2染色. 所以$\chi(G)\leq 2+3=5$. +\item[9.] 令$e=(u,v)$, 假设$G\setminus e$ 颜色最少的染色中 $u,v$颜色相同, 则该染色在$G/e$中也是可行的染色且$G/e$中任何可行的染色都得到$G\setminus e$ 中$u,v$颜色相同的一个染色. 假设$G\setminus e$ 颜色最少的染色中 $u,v$颜色不同, 同理任何一种染色对应着$G$的染色. 因此$\chi(G\setminus e)=\min(\chi(G),\chi(G/e))$. +\item[34.] $P_k(C_n)=P_k(P_n)-P_k(C_{n-1})=k(k-1)^{n-1}-P_k(C_{n-1})$ 展开得到$P(C_n,k)=(k-1)^n+(-1)^n(k-1)$. +\item[35.] $\sum_{i=0}^k[(i-1)^n+(-1)^n(i-1)](k-i)$ +\end{itemize} +\section*{第九章} +\begin{itemize} +\item[2.] +\item[5.] 1强联通. 2单向联通. 3.弱联通 4.单向联通 5.强联通 6.强联通 +\item[7.] 略 +\item[11.] 对子树(点数)做归纳. base case是只包含一个点的二元树,显然成立. 对$T$中的所有非叶子节点$u$, 存在一个以$u$为根的二元树, 其左右子树都满足性质$m=2t-2$, 因此$m'=m_1+m_2+2=2t_1+2t_2-4+2=2t-2$. \end{itemize} \end{document} \ No newline at end of file