Monochromatic Directed Walks in Arc-colored Directed Graphs

All graphs considered here are directed and have no loops. For a directed graph D, let V(D) denote the set of vertices of D, 2E(D) its set of arcs, and z(D) the chromatic number of D. D is symmetric iff (x, y)~E(D)~(y, x)EE(D). A directed walk of length k in D is a sequence o f k arcs (not necessarily distinct), e~, e2, ..., ek such that the initial vertex of ei+l is the terminal vertex of e, for i= 1, 2 . . . . , k 1. The directed walk above is called a directed path if all the k + 1 vertices incident with its arcs are distinct. An arc-coloring of D is a mapping of 2E(D) into a set C of colors. A subgraph of D is monochromatic if all its arcs have the same color. Gallai [5] and Roy [7] proved independently the first result connecting the chromatic number of a directed graph with the maximal length of a directed path in it; Every directed graph D contains a directed path of length z ( D ) I . Chwital [2] noticed that the result of Gallai and Roy implies the following extension of a result of Busolini [1]: