The Helly property is definitively needed. For instance, convex compact subsets of Rd have the Leray property, but not the Helly property. Surprisingly all graphs are intersection graphs of convex compact subsets of R3, and all octahedrons (but not all graphs) are intersection graphs of convex compact subsets of R2, compare [W67].
|What happens if only relaxed versions of the Leray-property are valid: Either we have 0= k= k+1 =... for every nonempty intersection of Sxs, for some fixed integer k, or we request k 0, 1, 2,... for every nonempty intersection of Sxs, again for some fixed k?|
Circular arcs of a cycle are an example where both the Helly property and the Leray property may be violated, but only slightly. The Helly property is violated if there are three arcs covering the whole circle, and the Leray property is violated if there are two arcs covering the whole circle. It quickly follows from the Theorem on intersection graphs of dual hypergraphs that 1(G^) =0 or 1 for every circular-arc graph G, depending on whether the arcs cover the whole circle and no three cover the whole circle.
|What can be said on the other Betti numbers of the complexes of circular-arc graphs?|
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