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Minimum vertex degree threshold for loose Hamilton cycles
 in 3graphs. Journal of Combinatorial Theory, Series B, accepted
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Minimum codegree threshold for hamilton cycles in kuniform hypergraphs
 Journal of Combinatorial Theory, Series A
, 2015
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On tours that contain all edges of a hypergraph
"... Let H be a kuniform hypergraph, k � 2. By an Euler tour in H we mean an alternating sequence v0,e1,v1,e2,v2,...,vm−1,em,vm = v0 of vertices and edges in H such that each edge of H appears in this sequence exactly once and vi−1,vi ∈ ei, vi−1 ̸ = vi, for every i = 1,2,...,m. This is an obvious genera ..."
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Let H be a kuniform hypergraph, k � 2. By an Euler tour in H we mean an alternating sequence v0,e1,v1,e2,v2,...,vm−1,em,vm = v0 of vertices and edges in H such that each edge of H appears in this sequence exactly once and vi−1,vi ∈ ei, vi−1 ̸ = vi, for every i = 1,2,...,m. This is an obvious generalization of the graph theoretic concept of an Euler tour. A straightforward necessary condition for existence of an Euler tour in a kuniform hypergraph is Vodd(H)  � (k − 2)E(H), where Vodd(H) is the set of vertices of odd degrees in H and E(H) is the set of edges in H. In this paper we show that this condition is also sufficient for hypergraphs of a broad class of kuniform hypergraphs, that we call strongly connected hypergraphs. This result reduces to the Euler theorem on existence of Euler tours, when k = 2, i.e. for graphs, and is quite simple to prove for k> 3. Therefore, we concentrate on the most interesting case of k = 3. In this case we further consider the problem of existence of an Euler tour in a certain class of 3uniform hypergraphs containing the class of strongly connected hypergraphs as a proper subclass. For hypergraphs in this class we give a sufficient condition for existence of an Euler tour and prove intractability (NPcompleteness) of the problem in this class in general.
Hamilton cycles in quasirandom hypergraphs
, 2015
"... We show that, for a natural notion of quasirandomness in kuniform hypergraphs, any quasirandom kuniform hypergraph on n vertices with constant edge density and minimum vertex degree Ω(nk−1) contains a loose Hamilton cycle. We also give a construction to show that a kuniform hypergraph satisfying ..."
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We show that, for a natural notion of quasirandomness in kuniform hypergraphs, any quasirandom kuniform hypergraph on n vertices with constant edge density and minimum vertex degree Ω(nk−1) contains a loose Hamilton cycle. We also give a construction to show that a kuniform hypergraph satisfying these conditions need not contain a Hamilton `cycle if k − ` divides k. The remaining values of ` form an interesting open question. 1
PERFECT MATCHINGS, TILINGS AND HAMILTON CYCLES IN HYPERGRAPHS
, 2015
"... This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite HajnalSzemerédi Theor ..."
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This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite HajnalSzemerédi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs. In particular, we determine the minimum codegree thresholds for Hamilton `cycles in large kuniform hypergraphs for ` < k/2. We also determine the minimum vertex degree threshold for loose Hamilton cycle in large 3uniform hypergraphs. These results generalize the wellknown theorem of Dirac for graphs. Third, we determine the minimum codegree threshold for near perfect matchings in large kuniform hypergraphs, thereby confirming a conjecture of Rödl, Ruciński and Szemerédi. We also show that the decision problem on whether a kuniform hypergraph with certain minimum codegree condition contains a perfect matching can be solved in polynomial time, which solves a problem of Karpiński, Ruciński and Szymańska completely. At last, we determine the minimum vertex degree threshold for perfect tilings of C34 in large 3uniform hypergraphs, where C34 is the unique 3uniform hypergraph on four vertices with two edges.