We show that if a line ` is an isolated line transversal to a finite familyF of (possibly intersecting) balls in R and no two balls are externally tangent on ` then there is a subfamily G ⊆ F of size at most 12 such that ` is an isolated line transversal to G. We generalize this result to families of semi-algebraic ovaloids.

We present a necessary and sufficient condition for a family of sets to possess a transversal. Its form follows that of P. Hall's theorem: a family has a transversal if and only if it does not contain one of a set of 'forbidden' substructures.

We give a simple and intuitive fixed parameter tractable algorithm for the Odd Cycle Transversal problem, running in time O(3 · k · |E| · |V |). Our algorithm is best viewed as a reinterpretation of the classical Iterative Compression algorithm for Odd Cycle Transversal by Reed, Smith and Vetta [8].

We suggest and explore a matroidal version of the Brualdi Ryser conjecture about Latin squares. We prove that any n × n matrix, whose rows and Columns are bases of a matroid, has an independent partial transversal of length d2n/3e. We show that for any n, there exists such a matrix with a maximal independent partial transversal of length at most n− 1.

In this paper, we consider another generalization for quasi-ideal orthodox transversal, the so-called S0-orthodox transversals. We give a structure theorem for regular semigroups with S0-orthodox transversals. If S0 is a S0-orthodox transversal of S then S can be described in terms of S0. Mathematics Subject Classification: 20M10

Transversal Design is a well known combinatorial design that has been used in deterministic key predistribution scheme. Merging of blocks in a design sometimes helps to obtain a key predistribution scheme with better performance. A deterministic merging strategy to merge the blocks has been discussed. Also, a simple key establishment method for transversal design based key predistribution schem...

Given an undirected graph G = (VG, EG) and a fixed pattern graph H = (VH , EH) with k vertices, we consider the H-Transversal and H-Packing problems. The former asks to find the smallest S ⊆ VG such that the subgraph induced by VG \ S does not have H as a subgraph, and the latter asks to find the maximum number of pairwise disjoint k-subsets S1, ..., Sm ⊆ VG such that the subgraph induced by ea...

A transversal design of order n and block size k, or TD(k; n), is a triple (X, G, B ), where X is a set of kn elements. G 5 hG1 , . . . , Gkj is a partition of X into k sets each of size n; each class of the partition is a group. B is a set of k-subsets of X, with the property that each B [ B satisfies uB > Giu 5 1 for each 1 # i # k; sets in B are blocks. Finally, each unordered pair of elemen...

The transversal number ~(H) of a hypergraph H is the minimum eardinality of a set of vertices that intersects all edges of H. For k ~ 1 define ck = supz(H)/(ra + n), where H ranges over all k-uniform hypergraphs with n vertices and m edges. Applying probabilistic arguments we show that ck = (1 + o ( 1 ) ) ~ , r . This settles a problem of Tuza. t ~