If the lines of the complete graph K,, are calmed so that no point is on more than +(n -1) lines of the same color or so that each point lies on more than $(5n + 8) lines of different colors, then K,, contains a cycle of length n with adjacent lines having different colors.
Let the lines of a graph G be colored. Let an AC,,, (of G) be a cycle of length m with adjacent lines having different colors.
Suppose K,, (the complete graph cln n points) is colored so that no point lies on more than A lines of the same color. Daykin asked in [2] whether K,, must then contain an AC,, for n sufficiently l;arge. He showed that it must for A = 2 and n 2 6. Bollobas and Erdas showed in [l] that there must be an AC,, for n > 69A.
In this paper we improve this resull; by reducing the bound here to n > 7A.
Bollobas and Erdiis also showed that if K,, is colored so every point lies on lines of 7n/8 different colors then K,, rnuf,t contain an AC,. We improve this result by showing that if K,, is colored so that every point lies on lines of more than (5n + 8)/7 different colors, then K,, must contain an AC,.
We also give a conjecture concerning a more general problem of this type.