 
Summary: JOURNAL OF COMBINATORIAL THEORY 38. 9394 (1985)
Note
Even Edge Colorings of a Graph
NOGA ALON
Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
AND
YOSHIMI EGAWA
Department of Applied Mathematics, Science University of Tokvo,
Shinjukuku. Tokyo, 162 Japan
Communicated by the Managing Editors
Received April 30, 1984
It is shown that the minimum number of colors needed to paint the edges of a
graph G so that in every cycle of G there is a nonzero even number of edges of at
least one color is rlog,X(G)l. (" 1985 Academic Press, Inc.
For a simple graph G, let E(G) denote the minimum number 1for which
there exists a partition of E(G) into 1 subsets Ej, 1d id 1, satisfying
for any cycle Z of G, 0 < [E(Z) n Eil= 0 (mod 2)
for at least one Ei. (1)
The number obtained by replacing the condition (1) by
