Heavy paths and cycles in weighted graphs

A weighted graph is a graph in which each edge e is assigned a non-negative\ud number $w(e)$, called the weight of $e$. In this paper, some theorems on the\ud existence of long paths and cycles in unweighted graphs are generalized to heavy\ud paths and cycles in weighted graphs

Directed paths with few or many colors in colored directed graphs

Given a graph $D=(V(D),A(D))$ and a coloring of $D$, not necessarily a proper coloring of either the arcs or the vertices of $D$, we consider the complexity of finding a path of $D$ from a given vertex $s$ to another given vertex $t$ with as few different colors as possible, and of finding one with as many different colors as possible. We show that the first problem is polynomial-time solvable, and that the second problem is NP-hard. \u