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...monstrate sharpness of some bounds. Our original proof of the lower bound was lengthy. The simple argument presented here originated with Dennis Eichhorn and Kevin O’Bryant. It will be generalized in =-=[7]-=- to compute edge-bandwidth for a large class of theta graphs. Example A. Consider G =Θ(l1,...,lm) with lm = 1 and l1 = ···= lm−1 = 3. Let ai,bi,ci denote the edges of the ith path of length 3, and let...

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...joint paths between them. A chorded cycle is an example of a θ-graph but, in general, a θ-graph need not be a chorded cycle. The idea of θ-graphs has been studied in a wide variety of situations (see =-=[2, 5, 8, 11, 15, 20, 22]-=-). In particular, every θ-graph contains an even cycle so, if a graph contains k disjoint θ-graphs, then it necessarily contains k disjoint even cycles. Hence, we define another function f(k, α). Let ...

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...′ � n � (Qn) ≤ 2 2 � n ⌊ n 2 ⌋ � − 1 (1) Recently, Tao Jiang el al. [13] rediscovered the edge-bandwidth for Kn, Kn,n and found an exact result for caterpillars. In a subsequent paper Eichhorn et al. =-=[5]-=- computed edge-bandwidths of all theta graphs. Let L(G) denote the line graph of G i.e. the graph whose vertices are edges of G and two vertices are adjacent if and only if the edges were incident in ...

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...ith |V (G) |≥ 3r+4s. Suppose for any pair of nonadjacent u and v in G, dG(u)+dG(v) ≥ 4r +6s− 1. Then G contains r + s disjoint cycles, such that s of them are chorded cycles. A theta graph is a the union of three internally disjoint paths that have the same two distinct end vertices. A chorded cycle is a simple example of a theta graph but, in general a theta graph needs not be a chorded cycle. It is obvious that K−4 is the theta graph with minimum order. In particular, every theta graph contains an even cycle, and the idea of theta graphs has been studied in a wide variety of situations (see [3, 6, 9, 11]). Our research is motivated by a classic extremal result, which obtained by Pósa and mentioned by Erdos in [7]. Note that it is also a basic fact that every graph G with order n ≥ 3 and size at least n contains a cycle. Theorem 1.2 [7] Every graph of order n ≥ 6 and size at least 3n−5 contains two disjoint cycles. Similarly, we are interested in the existence of disjoint theta graphs, since if a graph G contains specified number of disjoint theta graphs, then G also contains the same number of disjoint even cycles. For a graph F of order k and an integer n ≥ k, the extremal number ex(n; F ) ...

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...plete, and complete bipartite graphs in [10, 11]. ∗ Department of Mathematics, California State University, San Bernardino, CA 92407 1Edge-bandwidths are also known for caterpillars and theta graphs =-=[12, 13]-=-. A useful graph for determining the edge-bandwidth of a graph is L(G), the line graph of G, where V (L(G)) = E(G) and x, y are adjacent in L(G) if and only if x and y share a vertex in G. Then combin...

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...ions to extremal graph theory came in the form of providing best-known constructions of one sort or another. These contributions are mentioned several places in [M2], and also in [AFM] and [JMSW]. In =-=[EMOW]-=-, Mubayi, O’Bryant, West and I solve an edge-labeling problem on thetagraphs. If we define an edge-labeling f of a graph G to be an injection from E(G) to the integers, then the edge-bandwidth of G is...

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...admission control problem. The edge-bandwidth problem has been solved for only a few classes of graphs such as complete graph, complete bipartite graph with equal partites, caterpillar and theta graph=-=[5]-=-. This paper establishes the edge-bandwidth of the tensor product of a path with a path and a path with a cycle. Optimal edge-numberings to achieve each of these edge-bandwidths are provided. 2000 Mat...