Shinsei Tazawa

A series of search designs for 2m factorial designs of resolution V which permit search of one or two unknown extra three-factor interactions

In the absence of four-factor and higher order interactions, we present a series of search designs for 2m factorials (m≥6) which allow the search of at most k (=1,2) nonnegligible three-factor interactions, and the estimation of them along with the general mean, main effects and two-factor interactions. These designs are derived from balanced arrays of strength 6. In particular, the nonisomorphic weighted graphs with 4 vertices in which two distinct vertices are assigned with integer weight ω (1≤ω≤3), are useful in obtaining search designs for k=2. Furthermore, it is shown that a search design obtained for each m≥6 is of the minimum number of treatments among balanced arrays of strenth 6. By modifying the results for m≥6, we also present a search design for m=5 and k=2.

Enumeration of unlabelled bicolored graphs by degree parities

Abstract This paper gives an ordinary generating function for unlabelled bicolored graphs with a given number of odd vertices, where the cardinalities of the bipartite sets are equal. Moreover, the generating functions for the cardinality of each bipartite set from 1 to 8 are listed.

Enumeration of digraphs with given number of vertices of odd out-degree and vertices of odd in-degree

Abstract In a digraph, a vertex of odd out(in)-degree is called an odd out(in)-vertex. This paper will give the ordinary generating functions for labelled digraphs and unlabelled withgiven numbers of odd out-vertices and odd in-vertices.

Enumeration of Bipartite Self-Complementary Graphs

Let e(m, n), o(m, n), bsc(m, n) be the number of unlabelled bipartite graphs with an even number of edges whose partite sets have m vertices and n vertices, the number of those with an odd number of edges, and the number of unlabelled bipartite self-complementary graphs whose partite sets have m vertices and n vertices, respectively. Then this paper shows that the equality bsc(m, n) = e(m, n) − o(m, n) holds.

Enumeration of unlabelled graphs with specified degree parities

Abstract This paper gives a generating function for unlabelled graphs of order n . The coefficient of each monomial in this function shows the number of unlabelled graphs with given size and the number of odd vertices. Furthermore, the numerical examples are given for 1⩽ n ⩽9.

Enumeration of connected graphs with cut vertices

Abstract This paper is related to the enumeration of labelled graphs with k (k⩾2) cut vertices distributed only on one block and unlabelled graphs with one cut vertex. In this paper we give the exponential generating function for labelled connected graphs having exactly n blocks and the k (k⩾2) cut vertices distributed only on one block, and the cycle index sum for all unlabelled connected graphs with one cut vertex. The numerical tables are also given.

On separable self-complementary graphs

Abstract In this paper, we describe the structure of separable self-complementary graphs.

The number of 4-cycles in triangle-free oriented graphs

Abstract We determine the maximum number of oriented 4-cycles possible in triangle-free oriented graphs and we characterize the extremal graphs.

Mixed-Type Multiple-Valued Filing Scheme of Order One and Two

This paper will present a filing scheme for multiple-valued records. It is assumed that records are characterized by m attributes having n values each, and the query set consists of two kinds of queries, where one (called a first-order query) specifies a value of one attribute and another one (called a second-order query) specifies values of two attributes. The number of buckets to be organized and the redundancy of the file are less than those of an inverted file, provided the distribution of records is uniform. The number of pointer fields to be attached to the accession number stored in a bucket is considerably reduced in this scheme. Every first-order query can be answered by the access to one bucket. Every second-order query can be answered by the access to one bucket and the traversal of its contents without any set operations.