19 research outputs found

    The Hamilton-Waterloo Problem with even cycle lengths

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    The Hamilton-Waterloo Problem HWP(v;m,n;α,β)(v;m,n;\alpha,\beta) asks for a 2-factorization of the complete graph KvK_v or KvIK_v-I, the complete graph with the edges of a 1-factor removed, into α\alpha CmC_m-factors and β\beta CnC_n-factors, where 3m<n3 \leq m < n. In the case that mm and nn are both even, the problem has been solved except possibly when 1{α,β}1 \in \{\alpha,\beta\} or when α\alpha and β\beta are both odd, in which case necessarily v2(mod4)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;α,β)(v;2m,2n;\alpha,\beta) for odd α\alpha and β\beta whenever the obvious necessary conditions hold, except possibly if β=1\beta=1; β=3\beta=3 and gcd(m,n)=1\gcd(m,n)=1; α=1\alpha=1; or v=2mn/gcd(m,n)v=2mn/\gcd(m,n). This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above

    A complete solution to the infinite Oberwolfach problem

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    Let FF be a 22-regular graph of order vv. The Oberwolfach problem, OP(F)OP(F), asks for a 22-factorization of the complete graph on vv vertices in which each 22-factor is isomorphic to FF. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group GG. We will also consider the same problem in the more general contest of graphs FF that are spanning subgraphs of an infinite complete graph K\mathbb{K} and we provide a solution when FF is locally finite. Moreover, we characterize the infinite subgraphs LL of FF such that there exists a solution to OP(F)OP(F) containing a solution to OP(L)OP(L)

    On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity

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    The Hamilton-Waterloo problem asks for a decomposition of the complete graph of order v into r copies of a 2-factor F1 and s copies of a 2-factor F2 such that r+s = v−1 2 . If F1 consists of m-cycles and F2 consists of n cycles, we say that a solution to (m, n)- HWP(v; r, s) exists. The goal is to find a decomposition for every possible pair (r, s). In this paper, we show that for odd x and y, there is a solution to (2kx, y)-HWP(vm; r, s) if gcd(x, y) ≥ 3, m ≥ 3, and both x and y divide v, except possibly when 1 ∈ {r, s}

    A constructive solution to the Oberwolfach Problem with a large cycle

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    For every 22-regular graph FF of order vv, the Oberwolfach problem OP(F)OP(F) asks whether there is a 22-factorization of KvK_v (vv odd) or KvK_v minus a 11-factor (vv even) into copies of FF. Posed by Ringel in 1967 and extensively studied ever since, this problem is still open. In this paper we construct solutions to OP(F)OP(F) whenever FF contains a cycle of length greater than an explicit lower bound. Our constructions combine the amalgamation-detachment technique with methods aimed at building 22-factorizations with an automorphism group having a nearly-regular action on the vertex-set.Comment: 11 page

    Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles

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    It is conjectured that for every pair (,m)(\ell,m) of odd integers greater than 2 with m1  (mod)m \equiv 1\; \pmod{\ell}, there exists a cyclic two-factorization of KmK_{\ell m} having exactly (m1)/2(m-1)/2 factors of type m\ell^m and all the others of type mm^{\ell}. The authors prove the conjecture in the affirmative when 1  (mod4)\ell \equiv 1\; \pmod{4} and m2+1m \geq \ell^2 -\ell + 1.Comment: 31 page
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