Factorization of Graphs

PhD ThesisFor d 1; s 0; a (d; d + s)-graph is a graph whose degrees all lie in the interval fd; d + 1; :::; d + sg. For r 1; a 0; an (r; r+a)-factor of a graph G is a spanning (r; r+a)-subgraph of G. An (r; r+a)-factorization of a graph G is a decomposition of G into edge-disjoint (r; r + a)-factors. A graph is (r; r + a)-factorable if it has an (r; r + a)-factorization. For t 1, let (r; s; a; t) be the least integer such that, if d (r; s; a; t), then every (d; d + s)-simple graph G has an (r; r + a)-factorization into x (r; r + a)-factors for at least t di erent values of x. Then we show that, for r 3 odd and a 2 even, (r; s; a; t) = ( r tr+s+1 a + (t 1)r + 1 if t 2, or t = 1 and a < r + s + 1; r if t = 1 and a r + s + 1; Similarily, we have evaluated (r; s; a; t) for all other values of r; s; a and t. We call (r; s; a; t) the simple graph threshold number. A pseudograph is a graph where multiple edges and multiple loops are allowed. A loop counts two towards the degree of the vertex it is on. A multigraph here has no loops. For t 1, let (r; s; a; t) be the least integer such that, if d (r; s; a; t), then every (d; d+s)-pseudograph G has an (r; r+a)-factorization into x (r; r+a)-factors for at least t di erent values of x. We call (r; s; a; t) as the pseudograph threshold number. We have also evaluated (r; s; a; t) for all values of r, s, a and t. Note that for r 3 (r; 0; 1; 1) = 1 meaning that (r; 0; 1; 1) cannot be given a nite value. This study provides various generalisations of Petersen's theorem that \Every 2k-regular graph is 2- factorable".