The Hamilton-Waterloo problem with cycle sizes of different parity

Not scheduled
15m
UP FHS (Koper)

UP FHS

Koper

Titov trg 5,Koper

Speaker

Prof. Adrian Pastine (Universidad Nacional de San Luis)

Description

The Hamilton-Waterloo problem asks for a decomposition of the complete graph into $r$ copies of a 2-factor $F_{1}$ and $s$ copies of a 2-factor $F_{2}$ such that $r+s=\left\lfloor\frac{v-1}{2}\right\rfloor$. If $F_{1}$ consists of $m$-cycles and $F_{2}$ consists of $n$ cycles, then we call such a decomposition a $(m,n)-$HWP$(v;r,s)$. The goal is to find a decomposition for every possible pair $(r,s)$. This problem has been studied in great depth in the cases when $m$ and $n$ have the same parity, but there are few general results for the case of different parity. In this work, we use rings of polynomials of the form $\mathbb{Z}_{2^{n}}[x]/\left\langle x^2+x+1\right\rangle$ to show that for odd $x$ and $y$, there is a $(2^kx,y)-$HWP$(vm;r,s)$ if $\gcd(x,y)\geq 3$, $m\geq 3$, and both $x$ and $y$ divide $v$, except possibly when $1\in\{r,s\}$.

Primary author

Prof. Adrian Pastine (Universidad Nacional de San Luis)

Co-author

Melissa Keranen (Michigan Technological University)

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