Several arguments are presented which provide restrictions on the possible number of crossings in drawings of bipartite graphs. In particular it is shown that cr(K5,n)=4[1/2n][1/2(n−1)] and cr(K6,n)=6[1/2n][1/2(n−1)].
Several arguments are presented which provide restrictions on the possible number of crossings in drawings of bipartite graphs. In particular it is shown that cr(K5,n)=4[1/2n][1/2(n−1)] and cr(K6,n)=6[1/2n][1/2(n−1)].
The subject of each of the five sections of this paper is the planted plane trees discussed by Harary, Prins, and Tutte [7]. A description of the content of the present work is given in Section 1. Section 2 is devoted to a definition of plane trees in terms of finite sets and relations defined on them—we hope this definition will replace the topological concepts introduced in [7]. A one-to-one correspondence between the classes of isomorphic planted plane trees with n+2 vertices and the classes of isomorphic 3-valent planted plane trees with 2n+2 vertices is given in Section 3. Sections 4 and 5 deal with enumeration problems.
Pairs of non-isomorphic strong tournaments of orders 5 and 6 are given for which the subtournaments of orders 4 and 5, respectively, are pairwise isomorphic. Herefore, only pairs of orders 3 and 4 were known.
Let L be a finite geometric lattice of dimension n, and let w(k) denote the number of elements in L of rank k. Two theorems about the numbers w(k) are proved: first, w(k)≥w(1) for k=2,3,…,n−1. Second, w(k)=w(1) if and only if k=n−1 and L is modular. Several corollaries concerning the “matching” of points and dual points are derived from these results.
This note is a continuation of [2]; we describe here how to enumerate classes of isomorphic graded posets defined on a finite set. Perhaps the most interesting aspect of the results presented here is that the enumeration of these complex structures can be carried out in the algebra of formal power series.
Let G be a graph, G′ an embedding of G as a straight 1-complex in Rn, the real coordinate space of dimension n; let Φ be a group of transformations mapping Rn to itself. If for every automorphism α of G we can find a member of Φ mapping G′ onto itself in such a way that it induces α in G′, we say that G′ is a Φ-symmetric embedding of G. In particular this paper discusses conditions for the existence of such an embedding when Φ is the group of autohomeomorphisms of Rn or the group of invertible linear transformations in Rn, and the graph is the complete graph Km.
Non-separable graphs are enumerated, and also graphs without end-points. The basic enumeration tool is sums of cycle indices of automorphism groups.
For each d such that d-1 is prime, a d-valent graph of girth 6 having 2(d2−d+1) vertices is exhibited. The method also gives the trivalent graph of girth 8 and 30 vertices.
The present paper deals with an apparently hitherto untreated problem in the theory of restricted partitions: What is the number Tn(r) of distinct partitions of the composite integer nr that can be made by partwise addition of n—not necessarily distinct—partitions of r? The answer is given in the form of a finite series of binomial coefficients multiplied by certain integer coefficients which dependend only on r:
In general the non-vanishing ci(r) must be determined by direct calculation; in this paper we give them for all r≤11. Several other interpretations of Tn(r) are given, and some additional open questions concerning the interpretation of the results are discussed.
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