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- Title
On the Basis Number of the Strong Product of Theta Graphs with Cycles.
- Authors
Jaradat, M. M. M.; Janem, M. F.; Alawneh, A. J.
- Abstract
A basis B for the cycle space C(G) of a graph G is called a d-fold if each edge of G occurs in at most d of the cycles in the basis B. A basis B for the cycle space C(G) of a graph G is Smarandachely if each edge of G occurs in at least 2 of the cycles in B. The basis number, b(G), of a graph G is defined to be the least integer d such that there is a d-fold basis of the cycle space of G. MacLane [20] made a connection between the the number of occurrence of edges of a graph in its cycle bases and the planarity of a graph, which is related with parallel bundles on planar map geometries, a kind of Smarandache geometries. In fact, he proved that a graph G is planar if and only if b(G) ≤ 2. Jaradat [10] gave an upper bound of the basis number of the strong product of a graph with a bipartite graph in terms of the factors. In this work, we show that the basis number of the strong product of a theta graph with a cycle is either 3 or 4. Our result, improves Jaradat's upper bound in the case of specializing the factors by a theta graph and a cycle.
- Subjects
COMPUTERS in graph theory; SMARANDACHE function; SMARANDACHE notions; INTEGER programming; MATHEMATICAL programming; BIPARTITE graphs; ANALYTIC geometry of planes
- Publication
International Journal of Mathematical Combinatorics, 2008, Vol 3, p40
- ISSN
1937-1055
- Publication type
Article