 
Summary: Acta Math.Hung.
49 (12)(1987),163167.
MONOCHROMATIC DIRECTED WALKS
IN ARCCOLORED DIRECTED GRAPHS
N. ALON (Cambridge, USA)
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 ofk 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 arccoloring of D is a mapping of 2E(D) into a set C of colors. A sub
graph of D is monochromatic if all its arcs have the same color.
Gallai [5] and Roy [7] proved independently the first result connecting the chro
matic 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]:
Trr~OREMA (Chvfital). Let D be a directed graph and let k, r be positive integers
such that z(D)>k'; then in any arccoloring of D with r colors, D contains a mono
