On inequivalent balanced incomplete block designs, I

AbstractThe existence of at least two inequivalent balanced incomplete block designs is established for certain designs with λ=1 and block size m+1 where m is a prime power. An asymptotic result for the number of inequivalent solutions of such designs is also proved

Non-isomorphic solutions of some balanced incomplete block designs. I

AbstractIn this paper we develop a method for generating non-isomorphic solutions of balanced incomplete block designs belonging to the series of symmetric designs with parameters (4t+3, 2t+1, t) and to the series with parameters (4t+4, 8t+6, 4t+3, 2t+2, 2t+1). We also prove a result about the number of non-isomorphic solutions of these designs as the parameter t tends to infinity

Characterizations of 2-variegated graphs and of 3-variegated graphs

AbstractA graph is said to be k-variegated if its vertex set can be partitioned into k equal parts such that each vertex is adjacent to exactly one vertex from every other part not containing it. We prove that a graph G on 2n vertices is 2-variegated if and only if there exists a set S of n independent edges in G such that no cycle in G contains an odd number of edges from S. We also characterize 3-variegated graphs

Graphs derivable from L<SUB>3</SUB>(5) graphs

We show that corresponding to a latin square of order 5 with an orthogonal mate there is essentially a unique L3(5) graph G1. Similarly corresponding to a latin square of order 5 without an orthogonal mate there is a unique L3(5) graph G2. G1 is the only member in the weak-equivalence class containing it and G1 has a unique pseudo-(3, 6, 3) graph as an ascendant. The weak-equivalence class containing G2 contains precisely the complement of G2 and two other graphs which are complements of each other. G2 has exactly two nonisomorphic pseudo-(3, 6, 3) graphs as ascendants. Both the weak-equivalence classes are closed with respect to complementation and hence the other possible ascendants of G1 and G2 respectively are nothing but the complements of the corresponding pseudo-(3, 6, 3) ascendants mentioned above

Nonisomorphic solutions of pseudo-(3.5,2) and pseudo-(3,6,3) graphs

In this paper we indicate the existence of 7 mutually nonisomorphic pseudo-(3,6,3) graphs and that of 9 mutually nonisomorphic pseudo-(3,5,2) graphs. Further the graphs obtained have been classified according to their mutual relationships

Non-isomorphic solutions of some balanced incomplete block designs. I

In this paper we develop a method for generating non-isomorphic solutions of balanced incomplete block designs belonging to the series of symmetric designs with parameters (4t+3, 2t+1, t) and to the series with parameters (4t+4, 8t+6, 4t+3, 2t+2, 2t+1). We also prove a result about the number of non-isomorphic solutions of these designs as the parameter t tends to infinity

Seidel-equivalence in LB<SUB>3</SUB>(6) graphs

It is proved that the Seidel-equivalent of a LB<SUB>3</SUB>(6) graphG with respect to a partition (W<SUB>1</SUB>,W<SUB>2</SUB>) of its vertex set is a pseudo-LB<SUB>3</SUB>(6) graph if and only if W<SUB>1</SUB> consists of six vertices lying on a line of G