Bounded graphs

AbstractA graph G is called bounded if for every mapping ƒ:V(G)→N there exists a sequence (Cn)nϵN of naturals such that for every ray U=(u1,u2,u3,…) of G an n0 can be found with ƒf(ui)< ci for all i⩾n0. All locally finite connected graphs are bounded. Four basic types of unbounded graphs are given, and it is conjectured that one of these configurations must be present in every unbounded graph. This conjecture is proved for the graphs with rayless blocks and the graphs which do not contain an infinite system of disjoint rays