Coverings and matchings in r-partite hypergraphs

Ryser\u27s conjecture postulates that for r -partite hypergraphs, τ ≤ (r - 1)ν where τ is the covering number of the hypergraph and ν is the matching number. Although this conjecture has been open since the 1960s, researchers have resolved it for special cases such as for intersecting hypergraphs where r ≤ 5. In this article, we prove several results pertaining to matchings and coverings in r -partite intersecting hypergraphs. First, we prove that finding a minimum cardinality vertex cover for an r -partite intersecting hypergraph is NP-hard. Second, we note Ryser\u27s conjecture for intersecting hypergraphs is easily resolved if a given hypergraph does not contain a particular subhypergraph, which we call a “tornado.” We prove several bounds on the covering number of tornados. Finally, we prove the integrality gap for the standard integer linear programming formulation of the maximum cardinality r -partite hypergraph matching problem is at least r - k where k is the smallest positive integer such that r - k is a prime power

Vertex covers by monochromatic pieces - A survey of results and problems

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

Partitioning 3-colored complete graphs into three monochromatic cycles

We show in this paper that in every 3-coloring of the edges of Kn all but o(n) of its vertices can be partitioned into three monochromatic cycles. From this, using our earlier results, actually it follows that we can partition all the vertices into at most 17 monochromatic cycles, improving the best known bounds. If the colors of the three monochromatic cycles must be different then one can cover ( 3 4 − o(1))n vertices and this is close to best possible