The Hilton-Spencer Cycle Theorems Via Katona’s Shadow Intersection Theorem

Abstract

Abstract A family 𝒜 of sets is said to be intersecting if every two sets in 𝒜 intersect. An intersecting family is said to be trivial if its sets have a common element. A graph G is said to be r-EKR if at least one of the largest intersecting families of independent r-element sets of G is trivial. Let α (G) and ω (G) denote the independence number and the clique number of G, respectively. Hilton and Spencer recently showed that if G is the vertex-disjoint union of a cycle C raised to the power k and s cycles 1C, . . ., sC raised to the powers k1, . . ., ks, respectively, 1 ≤ r ≤ α (G), and min(ω(C1k1),…,ω(Csks))≥ω(Ck), \min \left( {\omega \left( {{}_1{C^{k1}}} \right), \ldots ,\omega \left( {{}_s{C^{ks}}} \right)} \right) \ge \omega \left( {{C^k}} \right), then G is r-EKR. They had shown that the same holds if C is replaced by a path P and the condition on the clique numbers is relaxed to min(ω(C1k1),…,ω(Csks))≥ω(Pk), \min \left( {\omega \left( {{}_1{C^{k1}}} \right), \ldots ,\omega \left( {{}_s{C^{ks}}} \right)} \right) \ge \omega \left( {{P^k}} \right), We use the classical Shadow Intersection Theorem of Katona to obtain a significantly shorter proof of each result for the case where the inequality for the minimum clique number is strict.