Periods in missing lengths of rainbow cycles

A cycle in an edge-colored graph is said to be rainbow if no two of its edges have the same color. For a complete, infinite, edge-colored graph G, define S(G) = {n ≥ 2 | no n-cycle of G is rainbow}. Then S(G) is a monoid with respect to the operation n ◦m = n + m − 2, and thus there is a least positive integer π(G), the period of S(G), such that S(G) contains the arithmetic progression {N + kπ(G) | k ≥ 0} for some sufficiently large N . Given that n ∈ S(G), what can be said about π(G)? Alexeev showed that π(G) = 1 when n ≥ 3 is odd, and conjectured that π(G) always divides 4. We prove Alexeev’s conjecture: Let p(n) = 1 when n is odd, p(n) = 2 when n is divisible by four, and p(n) = 4 otherwise. If 2 < n ∈ S(G) then π(G) is a divisor of p(n). Moreover, S(G) contains the arithmetic progression {N + kp(n) | k ≥ 0} for some N = O(n). The key observations are: If 2 < n = 2k ∈ S(G) then 3n− 8 ∈ S(G). If 16 6= n = 4k ∈ S(G) then 3n− 10 ∈ S(G). The main result cannot be improved since for every k > 0 there are G, H such that 4k ∈ S(G), π(G) = 2, and 4k + 2 ∈ S(H), π(H) = 4.