1,280 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

    ON GRAPH DECOMPOSITIONS AND DESIGNS: EXPLORING THE HAMILTON-WATERLOO PROBLEM WITH A FACTOR OF 6-CYCLES AND PROJECTIVE PLANES OF ORDER 16

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    This dissertation tackles the challenging graph decomposition problem of finding solutions to the uniform case of the Hamilton-Waterloo Problem (HWP). The HWP seeks decompositions of complete graphs into cycles of specific lengths. Here, we focus on cases with a single factor of 6-cycles. The dissertation then delves into the construction of 1-rotational designs, a concept from finite geometry. It explores the connection between these designs and finite projective planes, which are specific geometric structures. Finally, the dissertation proposes a potential link between these seemingly separate areas. It suggests investigating whether 1-rotational designs might hold the key to solving unsolved instances of the uniform HWP. By exploring this connection, the research aims to find new constructions and solutions for these long standing open problems

    On uniformly resolvable (C4,K1,3)(C_4,K_{1,3})-designs

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    In this paper we consider the uniformly resolvable decompositions of the complete graph KvK_v minus a 1-factor (KvI)(K_v − I) into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either 4-cycles or 3-stars

    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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