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- W2895064019 abstract "The Hamilton-Waterloo Problem HWP$(v;m,n;alpha,beta)$ asks for a 2-factorization of the complete graph $K_v$ or $K_v-I$, the complete graph with the edges of a 1-factor removed, into $alpha$ $C_m$-factors and $beta$ $C_n$-factors, where $3 leq m < n$. In the case that $m$ and $n$ are both even, the problem has been solved except possibly when $1 in {alpha,beta}$ or when $alpha$ and $beta$ are both odd, in which case necessarily $v equiv 2 pmod{4}$. In this paper, we develop a new construction that creates factorizations with larger cycles from existing factorizations under certain conditions. This construction enables us to show that there is a solution to HWP$(v;2m,2n;alpha,beta)$ for odd $alpha$ and $beta$ whenever the obvious necessary conditions hold, except possibly if $beta=1$; $beta=3$ and $gcd(m,n)=1$; $alpha=1$; or $v=2mn/gcd(m,n)$. This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above." @default.
- W2895064019 created "2018-10-12" @default.
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- W2895064019 date "2018-10-03" @default.
- W2895064019 modified "2023-09-25" @default.
- W2895064019 title "The Hamilton-Waterloo Problem with even cycle lengths" @default.
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- W2895064019 doi "https://doi.org/10.48550/arxiv.1810.02009" @default.
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