Materials Science & Technology
http://hdl.handle.net/10576/3401
2023-12-06T02:17:28ZThe basis number of the strong product of paths and cycles with bipartite graphs
http://hdl.handle.net/10576/49112
The basis number of the strong product of paths and cycles with bipartite graphs
Jaradat, M.M.M.
The basis number of a graph G is defined to be the least integer d such that there is a basis B of the cycle space of G such that each edge of G is contained in at most d members of B. MacLane [13] proved that a graph G is planar if and only if the basis number of G is less than or equal to 2. Ali [3] proved that the basis number of the strong product of a path and a star is less than or equal to 4. In this work, (1) We give an appropriate decomposition of trees. (2) We give an upper bound of the basis number of a cycle and a bipartite graph. (3) We give an upper bound of the basis number of a path and a bipartite graph. This is a generalization of Ali's result [3].
2007-01-01T00:00:00ZThe ramsey number for theta graph versus a clique of order three and four
http://hdl.handle.net/10576/49115
The ramsey number for theta graph versus a clique of order three and four
Bataineh, M.S.A.; Jaradat, M.M.M.; Bateeha, M.S.
For any two graphs F1 and F2, the graph Ramsey number r(F1, F2) is
the smallest positive integer N with the property that every graph on at
least N vertices contains F1 or its complement contains F2 as a subgraph.
In this paper, we consider the Ramsey numbers for theta-complete graphs.
We determine r(θn, Km) for m = 2, 3, 4 and n > m. More specifically, we
establish that r(θn, Km) = (n − 1)(m − 1) + 1 for m = 3, 4 and n > m.
2014-01-01T00:00:00ZThe cycle-complete graph Ramsey number r(C6,K8)≤38
http://hdl.handle.net/10576/49113
The cycle-complete graph Ramsey number r(C6,K8)≤38
Jaradat, M.M.M.; Alzaleq, B.M.N.
The cycle-complete graph Ramsey number
r
(
C
m
,
K
n
)
is the smallest integer
N
such that every graph
G
of order
N
contains a cycle
C
m
on m vertices or has independent number
α
(
G
)
≥
n
. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that
r
(
C
m
,
K
n
)=(
m
−
1
)
(
n
−
1
)
+
1
for all
m
≥
n
≥
3
(except
r
(
C
3
,
K
3
)
=
6
). In this paper, we show that
r
(
C
6
,
K
8
)
≤
38
.
2008-01-01T00:00:00ZThe cycle-complete graph Ramsey number
r
(
C
8
,
K
8
)
http://hdl.handle.net/10576/49114
The cycle-complete graph Ramsey number
r
(
C
8
,
K
8
)
Jaradat, M.M.M.; Alzaleq, B.M.N.
The cycle-complete graph Ramsey number
r
(
C
m
,
K
n
)
is the smallest integer
N
such that every graph
G
of order
N
contains a cycle
C
m
on
m
vertices or has independent number
α
(
G
)
≥
n
. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that
r
(
C
m
,
K
n
)
=
(
m
−
1
)
(
n
−
1
)
+
1
for all
m
≥
n
≥
3
(except
r
(
C
3
,
K
3
)
=
6
). In this paper we will present a proof for the conjecture in the case
n
=
m
=
8
.
2007-01-01T00:00:00Z