We develop the relationship between distributive lattices and stopping variable problems by showing that the class of stopping variables has this structure. Using representation theory for distributive lattices we reduce the “secretary problem” and the Sn/n, problem for Bernoulli trials to linear programming problems.
The primary result of this paper is the resolution of the question: Which non-Abelian discrete groups G satisfy (for some n>1) |(aS)n|=|Sn| for all S G and a∈G, (An) where |*| denotes the counting measure and Sn={s1…sn:si∈S, 1≤i≤n}?
We prove that a discrete group G satisfies (An) for some integer n>1 iff G is a finite Hamiltonian group. Furthermore, if γ denotes the invariant defined for finite Abelian groups introduced in [1] and H is any finite Hamiltonian group, then H satisfies (An) iff γ(H′)≥n, where H′ denotes the unique (up to isomorphism) maximal Abelian subgroup of H. In the course of this development a number of results concerning finite Hamiltonian groups are obtained. We conclude with a section on related conditions as well as a discussion of the general locally compact case.
We investigate Ford's method for generating a de Bruijn cycle of degree n. We show that the feedback function which generates the cycle has the minimum possible number of positions equal to 1 and we give an algorithm which finds those positions for all n.
Dilworth's theorem gives the number of chains whose union is a given partially ordered set in terms of intrinsic properties of the partial ordering. This paper uses Dilworth's theorem to find the number of chains needed to uniquely determine a given partially ordered set in terms of intrinsic properties of the partial ordering. If a covers b and c covers d, then (a, b) and (c, d) are incomparable covers if either a or b is incomparable with either c or d. We prove that the number of chains whose transtitive closure is a given partial ordering is the largest number of elements in any set of incomparable covers plus the number of isolated elements of the partially ordered set.
The smallest number of vertices, edges, or faces of any 3-polytope with no Hamiltonian circuit is determined. Similar results are found for simplicial polytopes with no Hamiltonian circuit.
Le Γ be the set of all finite graphs which are not embeddable into the projective plane. We write G1≺G2, if G2 is homomorphic to G1, that is, if there is a subgraph G′2 of G2 such that G1 can be obtained from G′2 by contraction of edges in G′2. The subset of all minimal graphs in the partially ordered set Γ≺ is called the minimal basis of Γ. We denote this basis by M(Γ). Every graph of M(Γ) is called a minimal graph (of Γ). In a former paper [2], we determined all disconnected minimal graphs and all minimal graphs separable by one or two vertices. Hence, we need only consider three-connected minimal graphs. Let us say that in our Proposition 7 of [2] we omitted a bracket in G14. In this graph, namely, the uppermost 2 must be replaced by <2>.
In the following we deal with two new cases, in which we succeed in determining the minimal graphs:
First, let us suppose that G∈M(Γ) has the properties: (1) G is (at least) three-connected and (2) there is a vertex a in G such that the graph G/a (that is, one has to destroy the vertex a and all edges of G incident to a) contains a topologic K5 but there exists no topologic K3,3 contained in G/a. We prove the theorem that there are exactly two graphs in M(Γ) satisfying (1) and (2). One of these graphs is the G14. The other graph, denoted by D, can be obtained from the K5 where we let T be a triple of neighboring edges in the K5, by inserting a new vertex on every edge of T and by then connecting these three new vertices through three edges with one further new vertex a lying outside the topologic K5 (see Figure 2).
Second, we consider non-planar graphs G with the property that G/a is planar for every vertex a of G. In another paper [3] these graphs have been called nearly planar. Our next question is: Are there nearly planar graphs not being embeddable into the projective plane? We prove the theorem that all nearly planar graphs can be embedded into the projective plane with a single exception of one graph F. This exceptional graph F consists of two disjoint K4 and four edges (ai, bi), i=1,…, 4 which join the vertices ai of the one K4 with the vertices bi of the other K4. We then show that this graph is a minimal graph of
A 5-dimensional convex polytope P is constructed whose graph G has the property that if it is the graph a convex polytope P′ then P′ is combinatorially equivalent to P and, furthermore, G can be realized as the graph of P in only one way (i.e., if a subgraph of G determines a face of P it also determines a face of P′).
If G is any finite Abelian group define
where ei are the canonic invariants of G (ei+1|ei for all i). The primary result is the “super-additivity” of γ, i.e.,
In the process of establishing (*) the structure of Abelian groups is studied in great detail and a technique is developed for proving inequalities analogous to (*) for other invariants than γ.
We then apply (*) to obtain various further results of a combinatorial nature.
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