The Generalized Almost Resolvable Cycle System Of Some Graphs

A k-cycle system of the undirected graph G with order n is a pair ((V(G),C), where C is a collection of edge disjoint k-cycles which partitions the edge set of G with vertex set V(G). Suppose n = kq + r where 0 < r < k. When r = 0, a collection of q = n/k vertex disjoint k-cycles is called a parallel class of C; the k-cycle system (V(G),C) is said to be resolvable if C can be partitioned into (n -1)/2 parallel classes. When r = 1, a collection of q = (n — 1)/k vertex disjoint k-cycles is called an almost parallel class of C; the k-cycle system (V(G),C) is said to be almost resolvable if C can be partitioned into almost parallel classes such that the remaining k-cycles are vertex disjoint. When 2 < r < k, a collection of q = (n — r)/k vertex disjoint k-cycles is called an almost parallel class of C; the k-cycle system (V(G),C) is said to be generalized almost resolvable if C can be partitioned into almost parallel classes such that the remaining k-cycles are vertex disjoint, and these remaining k-cycles are referred to as a short parallel class.This paper which consists of three parts mainly considers the generalized almost resolvable cycle system for some graphs.In the first chapter, the concepts and a short survey of generalized almost resolvable cycle system are given. Some notations, symbols and the main results are also given.In the second chapter, we generalize the result of paper [1] which considered the almost resolvable 2k-cycle systems of Kn for n' odd. This paper investigates general-ized almost resolvable 2k-cycle systems of Kn- I for n even. We show the spectrum of GARCS(2k, Kn - I)s where I is a 1-factor of Kn, when 2k = 4, 6, 10 and 14. We also show that the spectrum for generalized almost resolvable 2k-cycle systems of Kn — I is precisely all n?2 (mod 8k), except possibly 16k+ 2, where I is a 1-factor of Kn and n is even, if k? 4 and there exists an almost resolvable 2k-cycle system of K4k+1.In the third chapter, we construct an generalized almost resolvable 6-cycle system of graph G = Kn,n — Dr1,r2(n?2( mod 12)), where D(r1, r2) is a double star with 2 centers of stars K1,r1 and K1,r2 connected by an edge, here r1 and r1 are both multiples of 12 and r1+r2=n-2.