889 research outputs found

    Resolution of the Oberwolfach problem

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    The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of K2n+1K_{2n+1} into edge-disjoint copies of a given 22-factor. We show that this can be achieved for all large nn. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large nn.Comment: 28 page

    Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams

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    The idea of computing Matveev complexity by using Heegaard decompositions has been recently developed by two different approaches: the first one for closed 3-manifolds via crystallization theory, yielding the notion of Gem-Matveev complexity; the other one for compact orientable 3-manifolds via generalized Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this paper we extend to the non-orientable case the definition of modified Heegaard complexity and prove that for closed 3-manifolds Gem-Matveev complexity and modified Heegaard complexity coincide. Hence, they turn out to be useful different tools to compute the same upper bound for Matveev complexity.Comment: 12 pages; accepted for publication in Topology and Its Applications, volume containing Proceedings of Prague Toposym 201

    On the decomposition threshold of a given graph

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    We study the FF-decomposition threshold δF\delta_F for a given graph FF. Here an FF-decomposition of a graph GG is a collection of edge-disjoint copies of FF in GG which together cover every edge of GG. (Such an FF-decomposition can only exist if GG is FF-divisible, i.e. if e(F)e(G)e(F)\mid e(G) and each vertex degree of GG can be expressed as a linear combination of the vertex degrees of FF.) The FF-decomposition threshold δF\delta_F is the smallest value ensuring that an FF-divisible graph GG on nn vertices with δ(G)(δF+o(1))n\delta(G)\ge(\delta_F+o(1))n has an FF-decomposition. Our main results imply the following for a given graph FF, where δF\delta_F^\ast is the fractional version of δF\delta_F and χ:=χ(F)\chi:=\chi(F): (i) δFmax{δF,11/(χ+1)}\delta_F\le \max\{\delta_F^\ast,1-1/(\chi+1)\}; (ii) if χ5\chi\ge 5, then δF{δF,11/χ,11/(χ+1)}\delta_F\in\{\delta_F^{\ast},1-1/\chi,1-1/(\chi+1)\}; (iii) we determine δF\delta_F if FF is bipartite. In particular, (i) implies that δKr=δKr\delta_{K_r}=\delta^\ast_{K_r}. Our proof involves further developments of the recent `iterative' absorbing approach.Comment: Final version, to appear in the Journal of Combinatorial Theory, Series

    Completing Partial Packings of Bipartite Graphs

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    Given a bipartite graph HH and an integer nn, let f(n;H)f(n;H) be the smallest integer such that, any set of edge disjoint copies of HH on nn vertices, can be extended to an HH-design on at most n+f(n;H)n+f(n;H) vertices. We establish tight bounds for the growth of f(n;H)f(n;H) as nn \rightarrow \infty. In particular, we prove the conjecture of F\"uredi and Lehel \cite{FuLe} that f(n;H)=o(n)f(n;H) = o(n). This settles a long-standing open problem

    A note about complexity of lens spaces

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    Within crystallization theory, (Matveev's) complexity of a 3-manifold can be estimated by means of the combinatorial notion of GM-complexity. In this paper, we prove that the GM-complexity of any lens space L(p,q), with p greater than 2, is bounded by S(p,q)-3, where S(p,q) denotes the sum of all partial quotients in the expansion of q/p as a regular continued fraction. The above upper bound had been already established with regard to complexity; its sharpness was conjectured by Matveev himself and has been recently proved for some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a consequence, infinite classes of 3-manifolds turn out to exist, where complexity and GM-complexity coincide. Moreover, we present and briefly analyze results arising from crystallization catalogues up to order 32, which prompt us to conjecture, for any lens space L(p,q) with p greater than 2, the following relation: k(L(p,q)) = 5 + 2 c(L(p,q)), where c(M) denotes the complexity of a 3-manifold M and k(M)+1 is half the minimum order of a crystallization of M.Comment: 14 pages, 2 figures; v2: we improved the paper (changes in Proposition 10; Corollary 9 and Proposition 11 added) taking into account Theorem 2.6 of arxiv:1310.1991v1 which makes use of our Prop. 6(b) (arxiv:1309.5728v1). Minor changes have been done, too, in particular to make references more essentia

    Computing Matveev's complexity via crystallization theory: the boundary case

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    The notion of Gem-Matveev complexity has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base D2\mathbb D^2 and two exceptional fibers and, therefore, for all torus knot complements.Comment: 27 pages, 14 figure
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