 
Summary: Properly colored Hamilton cycles in edge colored complete graphs
N. Alon
G. Gutin
Dedicated to the memory of Paul Erdos
Abstract
It is shown that for every > 0 and n > n0( ), any complete graph K on n vertices whose
edges are colored so that no vertex is incident with more than (1  1
2
 )n edges of the same
color, contains a Hamilton cycle in which adjacent edges have distinct colors. Moreover, for every
k between 3 and n any such K contains a cycle of length k in which adjacent edges have distinct
colors.
1 Introduction
Let Gc denote a graph G whose edges are colored in an arbitrary way. In particular, Kc
n denotes
an edgecolored complete graph on n vertices and Kc
m,m denotes an edgecolored complete bipartite
graph with equal partite sets of cardinality m each. For an edgecolored graph Gc, let (Gc) denote
the maximum number of edges of the same color incident with a vertex of Gc. A properly colored
cycle in Gc, that is, a cycle in which adjacent edges have distinct colors is called an alternating cycle.
