6 research outputs found
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".