Decompositions Of K_ν Into κ-circles With One Chord

Let Kv be the complete graph with v vertices and G be a finite simple graph. A G-design of Ku, denoted by (v,G,l}-GD, is a pair (X.B), where .Y is the vertex set of Kv and B is a collection of subgraphs of Ku called blocks, such that each block is isomorphic to G and any two vertices in Ku are jointed in exactly one block of B.In this thesis, the discussed finite simple graphs are C(r)m, i.e., one circle of length m with one chord, where r is the number of vertices between the ends of the chord, 1 < r < [(m-2)/m\- We give a unified method to construct C(r)m-designs. and obtain a series of results for the existence of C(r)2k-1-GD(v). v = 0,1 (mod 4k) and C(r)2k-GD(v), v = 0,I (mod 2/c + l). In Chapter 2, two important recursion theorems reduce the infinite admissible orders to four orders for given k. Furthermore, for each k, about two-thirds of these orders (for 1 < r < k - 1 or 1 < r < k - 2) are completely solved in Chapter 3 and 4.A complete symmetric directed multigraph of order n, denoted by Dn, is a directed graph with n vertices and any two distinct vertices x and y are joined by A arcs (x,y) and arcs (y,x). A directed m-cycle system, of order n with index A is a collection B of directed m-cycles, such that each arc of Dn appears exactly A times in B. Sometimes, it is called Mendelsohn design and is denoted by MD(n,m, }. When = 1. it is denoted by m-MD(n) briefly.In Chapter 5, we give the existence spectrum of m-MD(n) for odd integer n.