Block intersection numbers of block designs

Block Intersection Numbers of Block Designs Ii

On the structure of 1-designs with at most two block intersection numbers

We introduce the notion of an unrefinable decomposition of a 1-design with at most two block intersection numbers, which is a certain decomposition of the 1-designs collection of blocks into other 1-designs. We discover an infinite family of 1-designs with at most two block intersection numbers that each have a unique unrefinable decomposition, and we give a polynomial-time algorithm to compute...

Silver Block Intersection Graphs of Steiner 2-Designs

For a block design D, a series of block intersection graphs Gi, or i-BIG(D), i = 0, . . . , k is defined in which the vertices are the blocks of D, with two vertices adjacent if and only if the corresponding blocks intersect in exactly i elements. A silver graph G is defined with respect to a maximum independent set of G, called an α-set. Let G be an r-regular graph and c be a proper (r + 1)-co...

Hamiltonicity and restricted block-intersection graphs of t-designs

Given a combinatorial designD with block set B, its traditional block-intersection graph GD is the graph having vertex set B such that two vertices b1 and b2 are adjacent if and only if b1 and b2 have non-empty intersection. In this paper we consider the S-block-intersection graph, in which two vertices b1 and b2 are adjacent if and only if |b1 ∩ b2| ∈ S. As our main result we prove that {1, 2,...

Cycles in the block-intersection graph of pairwise balanced designs

The focws of this thesis is to investigate certain graph theoretic propertir-s of a class of graphs \shicL arise from combinatorial design theory. The Itlock-i~~tc.rscc.tion graph of a pairwise balanced design has as its vertices tlw Mocks of tllc (hasign ;irici has as its edges precisely those pairs of blocks wliicll llrtvr. nori-cm~pty ilitc~rsc~c'ticm. These graphs have a particular local st...