Locally Pancyclic Graphs

We prove the following theorem. LetGbe a graph of ordernand letW⊆V(G). If |W|⩾3 anddG(x)+dG(y)⩾nfor every pair of non-adjacent verticesx, y∈W, then eitherGcontains cyclesC3, C4, …, C|W|such thatCicontains exactlyivertices fromW(i=3, 4, …, |W|), or |W|=nandG=Kn/2, n/2, or else |W|=4,G[W]=K2, 2. This generalizes a result of J. A. Bondy (1971,J. Combin. Theory,11, 80–84) who proved the above for |W|=n, and also a recent result of B. Bollobás and G. Brightwell (1993,Combinatorica,13, 147–155), ensuring the existence ofC|W|only.