For natural numbers \(n\) and \(d\), let \(K_n(\Delta_c \leq d)\) denote a complete graph of order \(n\) whose edges are colored so that no vertex belongs to more than \(d\) edges of the same color, and where \(\Delta_c\) is the maximal degree in the subgraph formed by the edges of color \(c\). D. E. Daykin proved that if \(d=2\) and \(n \geq 6\), then every such graph contains an alternating hamiltonian cycle (i.e. a spanning cycle whose adjacent edges have different colors). The authors have extended this as follows. Theorem: If \(69d