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- W2950233005 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)={nge 2;|;text{no $n$-cycle of $G$ is rainbow}}$. Then $mathfrak{S}(G)$ is a monoid with respect to the operation $ncirc m = n+m-2$, 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 ${N+kpi(G);|;kge 0}$ for some sufficiently large $N$. Given that $ninmathfrak{S}(G)$, what can be said about $pi(G)$? Alexeev showed that $pi(G)=1$ when $nge 3$ 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 $2<ninmathfrak{S}(G)$ then $pi(G)$ is a divisor of $p(n)$. Moreover, $mathfrak{S}(G)$ contains the arithmetic progression ${N+kp(n);|;kge 0}$ for some $N=O(n^2)$. The key observations are: If $2<n=2kinmathfrak{S}(G)$ then $3n-8inmathfrak{S}(G)$. If $16ne n=4kinmathfrak{S}(G)$ then $3n-10inmathfrak{S}(G)$. The main result cannot be improved since for every $k>0$ there are $G$, $H$ such that $4kinmathfrak{S}(G)$, $pi(G)=2$, and $4k+2inmathfrak{S}(H)$, $pi(H)=4$." @default.
- W2950233005 created "2019-06-27" @default.
- W2950233005 creator A5059692424 @default.
- W2950233005 date "2015-09-18" @default.
- W2950233005 modified "2023-09-27" @default.
- W2950233005 title "Periods in missing lengths of rainbow cycles" @default.
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