Abstract
Given 0 < p < 1, we prove that a pseudorandom graph G with edge density p and sufficiently large order has the following property: Consider any red/blue-coloring of the edges of G and let r denote the proportion of edges which have the color red. Then there is a Hamilton cycle C so that the proportion of red edges of C is close to r. The analogue also holds for perfect matchings instead of Hamilton cycles. We also prove a bipartite version which is used elsewhere to give a minimum-degree condition for the existence of a Hamilton cycle in a 3-uniform hypergraph.
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