Pan-path Of Digraph

Our work is mainly about the problem of pan-path in digraph. In a digraph with order n, u and v are a pair of vertices of pan-path, which means that there is a path whose length is k, and k = 1,2,…, n - 1. That the pan-path of digraph with order n means there is a path whose length is k, and k = 1,2,…,n-1.In Chapter one, first we introduce some relative terminologs and notationys. Sec-ond we give some proved results about cycle and path of digraph.In Chapter two, first we research pan-path of a pair of vertex in particular tour-nament, and have following conclusion:Theorem Let Tn be a tournament on n vertices and d+(u) = max{d+(x);x∈V(T)}, assume that the vertices of T[V(T)\{u}] are labeled u1,u2,…,un-1 such that d+(u1)≤d+(u2)≤…≤d+(un-1), let u, v be a pair of vertex of Tn which can be chosen as follows: (1)if d+(u) =n-1 then v is arbitrary (2)if d+(u)≤n-1, and T[V(T)\{u}] is strong tournament then d+(v)= min{d+(x); x∈V(T)\{u} and x is not a bridgehead}. Then u and v are a pair of vertex of pan-path.Theorem Let T be an tournament of order n withδ-(T)>0 and let 3 (?), then there exists a k-path with 5≤k≤n - 1 between a pair of vertices u, v in D.Theorem Every strong in-tournament is pan-panth...