Donovan R. Hare

Edge-pancyclic block-intersection graphs

Abstract It is shown that the block-intersection graph of both a balanced incomplete block design with block size at least 3 and λ = 1, and a transversal design is edge-pancyclic.

Cycles in the block-intersection graph of pairwise balanced designs

It is known that the block-intersection graph of a pairwise balance design with ?=1 is edge-pancyclic given that its minimum block cardinality is at least 3.

Sparse color-critical hypergraphs

In this paper we obtain estimates for the least number of edges ann-uniformr-color-critical hypergraph of orderm may have.

Arithmetic Progressions in Sequences with Bounded Gaps

LetG(k, r) denote the smallest positive integergsuch that if 1=a1, a2, ?, agis a strictly increasing sequence of integers with bounded gapsaj+1?aj?r, 1?j?g?1, then {a1, a2, ?, ag} contains ak-term arithmetic progression. It is shown thatG(k, 2)>(k?1)/2(43)(k?1)/2,G(k, 3)>(2k?2/ek)(1+o(1)),G(k, 2r?1)>(rk?2/ek)(1+o(1)),r?2.

The connectivity of the block-intersection graphs of designs

It is shown that the vertex connectivity of the block-intersection graph of a balanced incomplete block design,BIBD (v, k, 1), is equal to its minimum degree. A similar statement is proved for the edge connectivity of the block-intersection graph of a pairwise balanced design,PBD (v, K, 1). A partial result on the vertex connectivity of these graphs is also given. Minimal vertex and edge cuts for the corresponding graphs are characterized.

Color-critical graphs and hypergraphs with few edges and no short cycles

Abstract We give constructions of color-critical graphs and hypergraphs with no cycles of length 5 or shorter and with relatively few edges.

Vertex transversals that dominate

For any graph, there is a largest integer k such that given any partition of the vertex set with at most k elements in each class of the partition, there is transversal of the partition that is a dominating set in the graph. Some basic results about this parameter, the partition domination number, are obtained. In particular, it is shown that its value is 2 for the two-dimensional infinite grid, and that determining whether a given vertex partition admits a dominating transversal is NP-complete, even for a graph which is a simple path. The existence of various dominating transversals in certain partitions in regular graphs is studied as well.

Square critically 3-chromatic hypergraphs

Abstract A hypergraph is said to be square if the number of its vertices equals the number of its edges. It is said to be critically 3-chromatic, or 3-critical, if it has chromatic number 3, but every proper subgraph has a 2-coloring. We investigate a number of questions concerning square 3-critical hypergraphs.

LARGE FACES IN 4-CRITICAL PLANAR GRAPHS WITH MINIMUM DEGREE 4

AbstractWe prove that the size of the largest face of a 4-critical planar graph with δ≥4 is at most one half the number of its vertices. Letf(n) denote the maximum of the sizes of largest faces of all such graphs withn vertices (n sufficiently large). We present an infinite family of graphs that shows