On the Hamilton-Waterloo Problem with cycle lengths of distinct parities

2018/01/23 by Burgess, Andrea, Danziger, Peter, Traetta, Tommaso
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.07638

Abstract

Let K_v^* denote the complete graph K_v if v is odd and K_v-I, the complete graph with the edges of a 1-factor removed, if v is even. Given non-negative integers v, M, N, α, β, the Hamilton-Waterloo problem asks for a 2-factorization of K^*_v into α C_M-factors and β C_N-factors. Clearly, M,N≥ 3, M| v, N| v and α+β= \lfloor(v-1)/(2)\rfloor are necessary conditions. Very little is known on the case where M and N have different parities. In this paper, we make some progress on this case by showing, among other things, that the above necessary conditions are sufficient whenever M|N, v>6N>36M, and β≥ 3.

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