 
Summary: Properly colored subgraphs and rainbow subgraphs in edgecolorings
with local constraints
Noga Alon
Tao Jiang
Zevi Miller
Dan Pritikin §
July 10, 2002
Abstract
We consider a canonical Ramsey type problem. An edgecoloring of a graph is called mgood if
each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let
f(m, G) denote the smallest n such that every mgood edgecoloring of Kn yields a properly edge
colored copy of G, and let g(m, G) denote the smallest n such that every mgood edgecoloring of
Kn yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs
G = Kt, we have c1mt2
/ ln t f(m, Kt) c2mt2
, and c1mt3
/ ln t g(m, Kt) c2mt3
/ ln t,
where c1, c2, c1, c2 are absolute constants. We also give bounds on f(m, G) and g(m, G) for
general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and
