King Arthur and his knights with two round tables

Hilton, A. J. W.; Holroyd, F. C. and Spencer, C. L. (2010). King Arthur and his knights with two round tables. The Quarterly Journal of Mathematics, 62(3) pp. 625–635.



A graph G is r-starred if, for some $v \in V(G)$, a largest pairwise intersecting family of independent r-subsets of V(G) may be obtained by taking all such subsets containing v (the 'r-star' at v). Let G be the disjoint union of powers of cycles; Hilton and Spencer have studied the problem of determining the values of r for which G is r-starred. They conjectuerd that the property holds for all r, and made a weaker conjecture that this is so for the union of just two cycles. In this paper we prove the second conjectuer, showing also that if G is the unionj of several graphs, each a power of a cycle, then G is $\alpha$-starred (where $\alpha$ is the independence number of G), provided that there is a homomorphism from some component of G to each of the other components.

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