Sarada Herke

On Ryser's conjecture for linear intersecting multipartite hypergraphs

Ryser conjectured that ź ≤ ( r - 1 ) ź for r -partite hypergraphs, where ź is the covering number and ź is the matching number. We prove this conjecture for r ≤ 9 in the special case of linear intersecting hypergraphs, in other words where every pair of lines meets in exactly one vertex.Aharoni formulated a stronger version of Rysers conjecture which specified that each r -partite hypergraph should have a cover of size ( r - 1 ) ź of a particular form. We provide a counterexample to Aharonis conjecture with r = 13 and ź = 1 .We also report a number of computational results. For r = 7 , we find that there is no linear intersecting hypergraph that achieves the equality ź = r - 1 in Rysers conjecture, although non-linear examples are known. We exhibit intersecting non-linear examples achieving equality for r ź { 9 , 13 , 17 } . Also, we find that r = 8 is the smallest value of r for which there exists a linear intersecting r -partite hypergraph that achieves ź = r - 1 and is not isomorphic to a subhypergraph of a projective plane.

On Hamilton decompositions of infinite circulant graphs

The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}.

Hamilton path decompositions of complete multipartite graphs

We prove that a complete multipartite graph

Parity of sets of mutually orthogonal Latin squares

K

Radial trees

with

Broadcasts and domination in trees

n>1

Perfect 1-Factorisations of Circulants with Small Degree

vertices and

On the Possible Orders of a Basis for a Finite Cyclic Group

m

Perfect 1‐Factorizations of a Family of Cayley Graphs

edges can be decomposed into edge-disjoint Hamilton paths if and only if

Decompositions of complete 3-uniform hypergraphs into small 3-uniform hypergraphs

\frac m{n-1}