Ryser conjectured that τ (r − 1)ν for r-partite hypergraphs, where τ is the covering number and ν 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 Ryser's conjecture which specified that each r-partite hypergraph should have a cover of size (r − 1)ν of a particular form. We provide a counterexample to Aharoni's conjecture with r = 13 and ν = 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 τ = r − 1 in Ryser's conjecture, although non-linear examples are known. We exhibit intersecting non-linear examples achieving equality for r ∈ {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 τ = r − 1 and is not isomorphic to a subhypergraph of a projective plane.
A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant graphs that admit a perfect 1-factorisation and we solve the existence problem for a large family of 4-regular circulants. Results of computer searches for perfect 1-factorisations of 4-regular circulant graphs of orders up to 30 are provided and some problems are posed.
The natural infinite analog of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2 -valent infinite circulant graph has a two-wayinfinite Hamilton path, there exist many such graphs that do not have a decomposition into edge-disjoint twoway-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2 -valent connected circulant graph has a decomposition into edge-disjoint Hamilton cycles. We settle the problem of decomposing 2 -valent infinite circulant graphs into edge-disjoint twoway-infinite Hamilton paths for = 2, in many cases when = 3, and in many other cases including where the connection set is ±{1, 2, … , } or ±{1, 2, … , − 1, + 1}.
K E Y W O R D Scirculant graph, hamilton decomposition, infinite graph
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