Materials Science & Technology http://hdl.handle.net/10576/3401 2023-12-06T02:17:28Z The 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  proved that a graph G is planar if and only if the basis number of G is less than or equal to 2. Ali  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 . 2007-01-01T00:00:00Z The 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. 2014-01-01T00:00:00Z The 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 . 2008-01-01T00:00:00Z The 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