Uniformly Resolvable Decompositions Of K_v Into K₂ And K_1,3 Graphs

Let Kv be the complete graph of order v. A uniformly resolvable (K2, K1,3)-decompositions of Kv,denoted by (K2,K1,3)-URD(v; r,s),is a decomposition of Kv into a set of subgraphs which can be partitioned into r parallel classes containing only copies of K2 and s parallel classes containing only copies of K1,3,such that every point of Kv appears exactly once in some subgraphs of each parallel class.The necessary conditions for the existence of a (K2, K1,3)-URD(v; r,s) are v ? 0 (mod 2) when s = 0 and v ? s ? 0 (mod 4) when s > 0. S. Kiicukcifci et al. solved the existence of a (K2, Ki,3)-URD(v; r, s) with minimum number of 1-factors and with 14 possible exceptions in 2015.In this thesis, we use perfect (K1,3,?)-frames and perfect incomplete URDs to give some new constructions for (K2, K1,3)-URD(v; r, s). A (K1,3, ?)-frame of type gu is a K1,3-decomposition of a complete u-partite graph with u parts of size g into partial parallel classes each of which is a partition of the vertex set except for these vertices in one of the u groups.We completely solve the existence of a (K1,3, ?)-frame of type gu, and we also solve the existence of a (K2, K1,3)-URD(v; r, s) for any admissible param-eters v, r and s.