66 research outputs found

    Hamiltonicity and σ\sigma-hypergraphs

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    We define and study a special type of hypergraph. A σ\sigma-hypergraph H=H(n,r,qH= H(n,r,q \mid σ\sigma), where σ\sigma is a partition of rr, is an rr-uniform hypergraph having nqnq vertices partitioned into n n classes of qq vertices each. If the classes are denoted by V1V_1, V2V_2,...,VnV_n, then a subset KK of V(H)V(H) of size rr is an edge if the partition of rr formed by the non-zero cardinalities \mid KK \cap ViV_i \mid, 1in 1 \leq i \leq n, is σ\sigma. The non-empty intersections KK \cap ViV_i are called the parts of KK, and s(σ)s(\sigma) denotes the number of parts. We consider various types of cycles in hypergraphs such as Berge cycles and sharp cycles in which only consecutive edges have a nonempty intersection. We show that most σ\sigma-hypergraphs contain a Hamiltonian Berge cycle and that, for ns+1n \geq s+1 and qr(r1)q \geq r(r-1), a σ\sigma-hypergraph HH always contains a sharp Hamiltonian cycle. We also extend this result to kk-intersecting cycles

    Combinatorics

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    Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

    The random k-matching-free process

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    Let P\mathcal{P} be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty nn-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that P\mathcal{P} is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the kk-matching-free process, where P\mathcal{P} is the property of not containing a matching of size kk. We are able to analyse the behaviour of this process for a wide range of values of kk; in particular we prove that if k=o(n)k=o(n) or if n2k=o(n/logn)n-2k=o(\sqrt{n}/\log n) then this process is likely to terminate in a kk-matching-free graph with the maximum possible number of edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds on kk are essentially best possible, and we make a first step towards understanding the behaviour of the process in the intermediate regime

    Fast winning strategies in Avoider-Enforcer games

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    In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not {\em who} wins but rather {\em how fast} can one win. These type of problems were studied earlier for Maker-Breaker games; here we initiate their study for unbiased Avoider-Enforcer games played on the edge set of the complete graph KnK_n on nn vertices. For several games that are known to be an Enforcer's win, we estimate quite precisely the minimum number of moves Enforcer has to play in order to win. We consider the non-planarity game, the connectivity game and the non-bipartite game

    Combinatorics, Probability and Computing

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    The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. The workshop also emphasized connections between probabilistic combinatorics and discrete probability

    EUROCOMB 21 Book of extended abstracts

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    Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves

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    Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to mathematical structures writ large. We achieve this horizontal generality by adopting a categorial perspective which allows us to show that: (1) structured decompositions (a recent, abstract generalization of many graph decompositions) define Grothendieck topologies on categories of data (adhesive categories) and that (2) any computational problem which can be represented as a sheaf with respect to these topologies can be decided in linear time on classes of inputs which admit decompositions of bounded width and whose decomposition shapes have bounded feedback vertex number. This immediately leads to algorithms on objects of any C-set category; these include -- to name but a few examples -- structures such as: symmetric graphs, directed graphs, directed multigraphs, hypergraphs, directed hypergraphs, databases, simplicial complexes, circular port graphs and half-edge graphs. Thus we initiate the bridging of tools from sheaf theory, structural graph theory and parameterized complexity theory; we believe this to be a very fruitful approach for a general, algebraic theory of dynamic programming algorithms. Finally we pair our theoretical results with concrete implementations of our main algorithmic contribution in the AlgebraicJulia ecosystem.Comment: Revised and simplified notation and improved exposition. The companion code can be found here: https://github.com/AlgebraicJulia/StructuredDecompositions.j

    Hamiltonicity below Dirac's condition

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    Dirac's theorem (1952) is a classical result of graph theory, stating that an nn-vertex graph (n3n \geq 3) is Hamiltonian if every vertex has degree at least n/2n/2. Both the value n/2n/2 and the requirement for every vertex to have high degree are necessary for the theorem to hold. In this work we give efficient algorithms for determining Hamiltonicity when either of the two conditions are relaxed. More precisely, we show that the Hamiltonian cycle problem can be solved in time cknO(1)c^k \cdot n^{O(1)}, for some fixed constant cc, if at least nkn-k vertices have degree at least n/2n/2, or if all vertices have degree at least n/2kn/2-k. The running time is, in both cases, asymptotically optimal, under the exponential-time hypothesis (ETH). The results extend the range of tractability of the Hamiltonian cycle problem, showing that it is fixed-parameter tractable when parameterized below a natural bound. In addition, for the first parameterization we show that a kernel with O(k)O(k) vertices can be found in polynomial time
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