科研之友
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摘要
Given k > 2 and two k -graphs (k -uniform hypergraphs) F and H, an F-factor in H is a set of vertex disjoint copies of F that together covers the vertex set of H. Lenz and Mubayi studied the F -factor problems in quasi-random k -graphs with minimum degree 12(nk-1). In particular, they constructed a sequence of 1/8-dense quasi-random 3-graphs H(n) with minimum degree 12(n2) and minimum codegree 12(n) but with no K2,2,2-factor. We prove that if p > 1/8 and F is a 3-partite 3-graph with f vertices, then for sufficiently large n, all p -dense quasi-random 3-graphs of order n with minimum codegree 12(n) and f | n have F -factors. That is, 1/8 is the density threshold for ensuring all 3-partite 3-graphs F -factors in quasi-random 3-graphs given a minimum codegree condition 12(n). Moreover, we show that one can not replace the minimum codegree condition by a minimum vertex degree condition. In fact, we find that for any p is an element of (0, 1) and n > n0 , there exist p-dense quasi-random 3-graphs of order n with minimum degree 12(n2) having no K2,2,2-factor. In particular, we study the optimal density threshold of F-factors for each 3 -partite 3-graph F in quasi-random 3-graphs given a minimum codegree condition 12(n). In addition, we also study F-factor problems for k-partite k-graphs F with stronger quasi-random assumption and minimum degree 12(nk-1).
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单位北京理工大学; 山东大学
