Fractional Matching Number Research Articles

Envy-free two-player [formula omitted]-cake and three-player two-cake divisions

Cloutier, Nyman, and Su (2005) initiated the study of envy-free cake-cutting problems involving several cakes. They showed that when there are two players and two or three cakes it is possible to find envy-free cake-divisions requiring few cuts, under natural assumptions.We prove that such a result also exists when there are two players and any number of cakes and when there are three players and two cakes. The proof relies on the fractional matching number in m-partite hypergraphs.

Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels

In this paper we study degree conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erdős on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum d-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also discuss an application to a problem of finding an optimal data allocation in a distributed storage system.

The fractional matching numbers of graphs

AbstractA fractional matching of a graph G is a function f that assigns to each edge a number in [0, 1] such that, for each vertex v, ∑ f(e) ≤ 1, where the sum is taken over all edges incident to v. The fractional matching number of G is the supremum of ∑e∈E(G) f(e) over all fractional matchings f. In this paper, we provide a new formula for calculating the fractional matching numbers of graphs using the Gallai–Edmonds Structure Theorem. Thus, we characterize graphs for which the fractional matching number equals the matching number and graphs for which the fractional matching number is the maximum possible (one‐half the number of vertices). © 2002 Wiley Periodicals, Inc.