### FOR THE

, 2011

"... A kayak paddle is a graph made of two cycles joined by a path. We can define KP (r, s, l) as two cycles of lengths r and s joined by a path of length l. If a graph G has m vertices and n edges, then a general vertex labeling of the graph is a one-to-one mapping of the vertex set of G into the set of ..."

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-labeling, then it can cyclically decompose K2n+1. D.

*Froncek*and L. Tollefson proved results for kayak paddles decomposing a complete graph using rosy labelings. In this thesis we investigate the existence of graceful labelings of kayak paddles.### DECOMPOSITION OF COMPLETE GRAPHS INTO SMALL GRAPHS

"... Abstract. In 1967, A. Rosa proved that if a bipartite graph G with n edges has an α-labeling, then for any positive integer p the complete graph K2np+1 can be cyclically decomposed into copies of G. This has become a part of graph theory folklore since then. In this note we prove a generalization of ..."

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Abstract. In 1967, A. Rosa proved that if a bipartite graph G with n edges has an α-labeling, then for any positive integer p the complete graph K2np+1 can be cyclically decomposed into copies of G. This has become a part of graph theory folklore since then. In this note we prove a generalization of this result. We show that every bipartite graph H which decomposes Kk and Km also decomposes Kkm.

### PRODUCT ROSY LABELING OF GRAPHS

, 2007

"... Abstract. In this paper we describe a natural extension of the well-known rosy labeling of graphs (also known as ρ-labeling). The labeling, called product rosy labeling, labels vertices with elements of products of additive groups. We illustrate the usefulness of this labeling by presenting a recurs ..."

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Abstract. In this paper we describe a natural extension of the well-known rosy labeling of graphs (also known as ρ-labeling). The labeling, called product rosy labeling, labels vertices with elements of products of additive groups. We illustrate the usefulness of this labeling by presenting a recursive construction of infinite families of trees decomposing complete graphs. 1.

### Super edge-graceful paths

, 2008

"... A graph G(V,E) of order |V | = p and size |E | = q is called super edge-graceful if there is a bijection f from E to {0, ±1, ±2,..., ± q−1 2} when q is odd and from E to {±1, ±2,..., ± q 2} when q is even such that the induced vertex labeling f ∗ defined by f ∗ (x) = ∑ xy∈E(G) f(xy) over all ed ..."

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A graph G(V,E) of order |V | = p and size |E | = q is called super edge-graceful if there is a bijection f from E to {0, ±1, ±2,..., ± q−1 2} when q is odd and from E to {±1, ±2,..., ± q 2} when q is even such that the induced vertex labeling f ∗ defined by f ∗ (x) = ∑ xy∈E(G) f(xy) over all edges xy is a bijection from V to {0, ±1, ±2..., ± p−1 2} when p is odd and from V to {±1, ±2,..., ± p} when p is even. 2 We prove that all paths Pn except P2 and P4 are super edgegraceful. 1

### Research Article A Lot-Size Model for Deteriorating Items under Conditions of a One-Time Only Extended Credit Period

"... Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Now-a-days, the offer of credit period to the customer for settling the account for the units purchased by the supplier is considered to be the mos ..."

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Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Now-a-days, the offer of credit period to the customer for settling the account for the units purchased by the supplier is considered to be the most beneficial policy. In this article, an attempt is made to formulate the mathematical model for a customer to determine optimal special cycle time when the supplier offers the special extended credit period for one time only during a special period. A decision policy for a retailer is developed to find optimal special cycle time. The theoretical results and effects of various parameters are studied by appropriate dataset. 1.

### Volume 2007, Article ID 46851, 8 pages doi:10.1155/2007/46851 Research Article

, 2007

"... A digraph is nonderogatory if its characteristic polynomial and minimal polynomial are equal. We find a characterization of nonderogatory unicyclic digraphs in terms of Hamil-tonicity conditions. An immediate consequence of this characterization ia that the com-plete product of difans and diwheels i ..."

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A digraph is nonderogatory if its characteristic polynomial and minimal polynomial are equal. We find a characterization of nonderogatory unicyclic digraphs in terms of Hamil-tonicity conditions. An immediate consequence of this characterization ia that the com-plete product of difans and diwheels is nonderogatory. Copyright © 2007 D. Bravo and J. Rada. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.

### Pancomponented 2-factorizations of complete graphs.

"... We pose and solve the existence of 2-factorizations of complete graphs and complete bipartite graphs that have the number of cycles per 2-factor varying, called pancomponented. Such 2-factorizations exist for all such graphs. The pancomponented problem requires a slight generalization of the methods ..."

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We pose and solve the existence of 2-factorizations of complete graphs and complete bipartite graphs that have the number of cycles per 2-factor varying, called pancomponented. Such 2-factorizations exist for all such graphs. The pancomponented problem requires a slight generalization of the methods used to solve pancyclic 2-factorization problem, by building 2-factors from cyclically generated 1-factors. These two solutions are offered as the basic approaches to constructing the two essential parameters of a 2-factorization: the size and the number of cycles in the

### doi:10.1155/2008/790964 Research Article Weak Forms of Continuity and Associated Properties

"... We introduce slightly p-continuous mapping and almost p-open mapping and investigate the relationships between these mappings and related types of mappings, and also study some properties of these mappings. Copyright q 2008 Mohammad S. Sarsak. This is an open access article distributed under the Cre ..."

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We introduce slightly p-continuous mapping and almost p-open mapping and investigate the relationships between these mappings and related types of mappings, and also study some properties of these mappings. Copyright q 2008 Mohammad S. Sarsak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and