Periods in missing lengths of rainbow cycles

TL;DR

This paper proves a conjecture about the periodic structure of missing rainbow cycle lengths in infinite edge-colored graphs, showing that the period divides specific integers depending on the cycle length's divisibility properties.

Contribution

It establishes that the period of missing rainbow cycle lengths divides certain integers based on the cycle length's parity and divisibility, confirming a conjecture and providing bounds for the arithmetic progressions.

Findings

The period divides p(n) depending on n's divisibility.

Missing cycle lengths form arithmetic progressions with step p(n).

Constructs examples with specific periods and missing lengths.

Abstract

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 $\mathfrak{S}(G)=\{n\ge 2\;|\;\text{no $n$-cycle of $G$ is rainbow}\}$. Then $\mathfrak{S}(G)$ is a monoid with respect to the operation $n\circ m = n+m-2$, and thus there is a least positive integer $\pi(G)$, the period of $\mathfrak{S}(G)$, such that $\mathfrak{S}(G)$ contains the arithmetic progression $\{N+k\pi(G)\;|\;k\ge 0\}$ for some sufficiently large $N$. Given that $n\in\mathfrak{S}(G)$, what can be said about $\pi(G)$? Alexeev showed that $\pi(G)=1$ when $n\ge 3$ is odd, and conjectured that $\pi(G)$ 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 $2<n\in\mathfrak{S}(G)$ then $\pi(G)$ is a divisor of $p(n)$. Moreover, $\mathfrak{S}(G)$ contains the arithmetic progression $\{N+kp(n)\;|\;k\ge 0\}$ for some $N=O(n^2)$. The key observations are: If $2<n=2k\in\mathfrak{S}(G)$ then $3n-8\in\mathfrak{S}(G)$. If $16\ne n=4k\in\mathfrak{S}(G)$ then $3n-10\in\mathfrak{S}(G)$. The main result cannot be improved since for every $k>0$ there are $G$, $H$ such that $4k\in\mathfrak{S}(G)$, $\pi(G)=2$, and $4k+2\in\mathfrak{S}(H)$, $\pi(H)=4$.