A graph-theoretical generalization of Berge’s analogue of the Erdős-Ko-Rado theorem

A family A of r-subsets of the vertex set V (G) of a graph G is intersecting if any two of the r-subsets have a non-empty intersection. The graph G is r-EKR if a largest intersecting family A of independent r-subsets of V (G) may be obtained by taking all independent r-subsets containing some particular vertex. In this paper, we show that if G consists of one path P raised to the power k0 ≥ 1, and s cycles 1C, 2C, . . . , sC raised to the powers k1, k2, . . . , ks respectively, with min ω(1C 1), ω(2C 2), . . . , ω(sC s) ≥ ω(P 0) ≥ 2 where ω(H) denotes the clique number of H, and if G has an independent r-set (so r is not too large), then G is r-EKR. An intersecting family of the largest possible size may be found by taking all independent r-subsets of V (G) containing one of the end-vertices of the path.