KPC2020(F未做)
E
*结论1**:\(\exists u\),\(\forall v((u,v)\in E)\),均满足\(d_v>d_u\)。则无解
结论1为无解的必要条件,下面通过构造证明其为充要条件
令\(f(u)\)为与\(u\)相邻的,\(d_x\)最小的任意一点
\(\forall u\),除\((f(u),u)\)外的边均设为\(\infty\),即其没有作用了
令\(x=f(u)\)
- \(d_x<d_u\),令\(dis(u,x)=d_u-d_x\),\(x,u\)同色
- \(d_x=d_u\),令\(dis(u,x)=d_x\),\(x,u\)异色
其实就是若干棵树,每棵树是一个根节点下面挂一个菊花