Helly-type theorems about sets
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Date
1980-02
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Discrete Mathematics
Abstract
Suppose that G is a graph. A 1-factor is a set of edges of G such that every vertex of G meets exactly one of its edges. Suppose that we have a set Y of 1-factors of G such that any two 1-factors of Y have an edge in common. We investigate the following questions: 1. (1) How large may Y be? 2. (2) When is there necessarily an edge contained in all the members of Y? We answer these questions in the case that G is the complete graph on 2n vertices K2n or the complete bipartite graph Kn, n. In the next section we study the first question; the third section is devoted to the second. In the final section we show that B2(Kn,n) ≈ Kn,n and B2(K2n) ≈ K2n. We end with some unsolved problems. In the remainder of this section we identify the 1-factors of the two graphs, state our results, and recall the definitions of the space of colorings of G. B(G). (See [4]. © 1980.