The intersection problem for a five vertex dragon

The intersection problem for block a..sks for values of k it is to find two B d and based on the V, with IBI n B:;d k; that is, k common blocks. We define to be the of cornDI.ete graph solve the intersection 1 1.1 Let G be a (undirected) which is some subgraph of another graph H. A G-decomposdion of H is a block (V, where V is the vertex set of Hand B decomposition of H into copies of G. In the case where H is Knl on n vertices, this is also called a G-design of order n. The intersection problem for asks for what values of k it is possible to find two (V, B1 ) and (V, two decompositions of a complete with vertex set V) with IBI n B21 k; that is, having precisely k common blocks. Define IG(Kn) to be the of achievable intersection sizes for a G-design of order n. It is easy to see that any two (V, Bd, (V, on the same vertex set with IBll IB21 = b blocks can never have b 1 blocks in common; that is, b 1 t/:. Ie;( Kn) for any G. This problenl has been considered for many combinatorial structures; a recent survey is [2]. In the specific case of small G-designs, where G is a graph with up to four vertices or up to four edges, a summary appears in [3]. Australasian Journal of Combinatorics (1996), pp. 109-120 For the rest of this paper we look ae(~onl1.p()S11tlOl:lS of comes from ;Cl,,~:pr1r.p(j that "this is to further paper by the latter paper does not mention the word. For the sake of having U'-'iUvU~.U.un the the been The denoted four to note that the vertex the free end of the 1{"",'r,('""" which form the 4. of J{n wi th vertex set then we denote the it a dragon of order n or e] or and The of this paper solution the intersection n"f",n'c> ..... for this fivevertex We show that for all n for which the set of intersection numbers is l.e. the number of blocks in tll(; 1.2 In order to find intersection Uu.uu,,, .... ,-,. two '''-'',iU' . .L'4 '-L' .• '" are trading. The former involves permuting the vertices of the original (J" D, where (J" is the D design. The latter (also referred to in the literature as mutually namely 'YlP'''''YYl'IIT1'Yl and