( r , r + 1 ) -factorizations of ( d , d + 1 ) -graphs

• Published in: Discrete mathematics Vol. 308; no. 5; pp. 645 - 669
• Main Author: Hilton, A.J.W.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 28-03-2008; Elsevier
• Subjects: Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Factor; Factorization; Graph; Information retrieval. Graph; Regular; Semiregular; Theoretical computing; Graph; Factorization; Regular; Semiregular; Factor; Graph decomposition; Graph degree; Multigraph; Regular graph
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2007.07.052

A
(
d
,
d
+
1
)
-graph is a graph whose vertices all have degrees in the set
{
d
,
d
+
1
}
. Such a graph is
semiregular. An
(
r
,
r
+
1
)
-factorization of a graph
G is a decomposition of
G into
(
r
,
r
+
1
)
-factors. For
d-regular simple graphs
G we say for which
x and
r G must have an
(
r
,
r
+
1
)
-factorization with exactly
x
(
r
,
r
+
1
)
-factors. We give similar results for
(
d
,
d
+
1
)
-simple graphs and for
(
d
,
d
+
1
)
-pseudographs. We also show that if
d
≥
2
r
2
+
3
r
-
1
, then any
(
d
,
d
+
1
)
-multigraph (without loops) has an
(
r
,
r
+
1
)
-factorization, and we give some information as to the number of
(
r
,
r
+
1
)
-factors which can be found in an
(
r
,
r
+
1
)
-factorization.