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Edgebandwidth of graphs
 SIAM J. Discrete Math
, 1999
"... Abstract. The edgebandwidth of a graph is the minimum, over all labelings of the edges with distinct integers, of the maximum difference between labels of two incident edges. We prove that edgebandwidth is at least as large as bandwidth for every graph, with equality for certain caterpillars. We o ..."
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Abstract. The edgebandwidth of a graph is the minimum, over all labelings of the edges with distinct integers, of the maximum difference between labels of two incident edges. We prove that edgebandwidth is at least as large as bandwidth for every graph, with equality for certain caterpillars. We obtain sharp or nearlysharp bounds on the change in edgebandwidth under addition, subdivision, or contraction of edges. We compute edgebandwidth for Kn, Kn,n, caterpillars, and some theta graphs. 1.
Independence number and disjoint theta graphs
, 2011
"... The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a θgraph to be a pair of vertices u, v with three internally disjoint paths joining u to v. Given an independence number α and a fixed integer k, the results contained in this paper provide sharp bound ..."
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The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a θgraph to be a pair of vertices u, v with three internally disjoint paths joining u to v. Given an independence number α and a fixed integer k, the results contained in this paper provide sharp bounds on the order f(k, α) of a graph with independence number α(G) ≤ α which contains no k disjoint θgraphs. Since every θgraph contains an even cycle, these results provide k disjoint even cycles in graphs of order at least f(k, α) + 1. We also discuss the relationship between this problem and a generalized ramsey problem involving sets of graphs.
Edgebandwidth of grids and tori
, 2005
"... The edgebandwidth of a graph G is the smallest number B0 for which there is an injective labeling of E(G) with integers such that the difference between the labels at any adjacent edges is at most B0. Here we compute the edgebandwidth for rectangular grids: B0(Pm \Phi Pn) = 2 min(m, n) 1, if ma ..."
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The edgebandwidth of a graph G is the smallest number B0 for which there is an injective labeling of E(G) with integers such that the difference between the labels at any adjacent edges is at most B0. Here we compute the edgebandwidth for rectangular grids: B0(Pm \Phi Pn) = 2 min(m, n) 1, if max(m, n)> = 3, where \Phi is the Cartesian product and Pn denotes the path on n vertices. This settles a conjecture of Calamoneri, Massini and Vr^to [Theoret. Computer Science, 307 (2003) 503 513]. We also compute the edgebandwidth of any torus (a product of two cycles) within an additive error of 5.
New results on edgebandwidth
 Theoret. Comput. Sci
"... The edgebandwidth problem is an analog of the classical bandwidth problem, in which one has to label the edges of a graph by distinct integers such that the maximum difference of labels of any two incident edges is minimized. We prove tight bounds on the edgebandwidth of hypercube and butterfly gr ..."
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The edgebandwidth problem is an analog of the classical bandwidth problem, in which one has to label the edges of a graph by distinct integers such that the maximum difference of labels of any two incident edges is minimized. We prove tight bounds on the edgebandwidth of hypercube and butterfly graphs and complete kary trees which extend and improve on previous known results. We also provide an improvement on the upper bound for the bandwidth of butterfly. 1
The Extremal Function for Two Disjoint Cycles *
"... Abstract A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order n ≥ 9 and size at least 7n−13 2 contains two disjoint theta graphs. We also show that every 2edgeconnected graph of order n ≥ 6 and size at least ..."
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Abstract A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order n ≥ 9 and size at least 7n−13 2 contains two disjoint theta graphs. We also show that every 2edgeconnected graph of order n ≥ 6 and size at least 3n − 5 contains two disjoint cycles, such that any specified vertex with degree at least three belongs to one of them. The lower bound on size in both are sharp in general.
Embedding of Cycles into Hypercube
 INTERNATIONAL JOURNAL OF INNOVATIONS IN ENGINEERING AND TECHNOLOGY (IJIET)
, 2012
"... We present an approach to find the edge congestion sum and dilation sum for embedding of cycle on n vertices, into hypercube. ..."
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We present an approach to find the edge congestion sum and dilation sum for embedding of cycle on n vertices, into hypercube.
EdgeBandwidth
, 2006
"... In this paper we discuss the edgebandwidth of some families of graphs and characterize graphs by edgebandwidth. In particular, bounds for m × n grids, triangular grids of size l, and the closure of the triangular grid T ∗ l. 1 ..."
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In this paper we discuss the edgebandwidth of some families of graphs and characterize graphs by edgebandwidth. In particular, bounds for m × n grids, triangular grids of size l, and the closure of the triangular grid T ∗ l. 1
RESEARCH STATEMENT
"... As a professional mathematician, the majority of my work has been in the areas of partition theory, analytic number theory, generalized Ramsey theory, and extremal graph theory. In the statement below, I will outline some of my contributions in these areas, and try to briefly explain the significanc ..."
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As a professional mathematician, the majority of my work has been in the areas of partition theory, analytic number theory, generalized Ramsey theory, and extremal graph theory. In the statement below, I will outline some of my contributions in these areas, and try to briefly explain the significance of my work by placing it in historical context. The study of partitions dates back over 300 years [D], and really began as an area of study with Euler [Eu]. Although it has venerable roots, partition theory still plays an important role at the cutting edge of mathematics today. First and foremost, partition theory holds some of the most beautiful results and fascinating open problems in the fields of additive number theory and enumerative combinatorics, some of which I will mention below. In addition to being an important part of combinatorics and additive number theory, there are links between partitions and Gauss ’ class number problem [OS], representation theory [JK], modular forms, elliptic curves [O2], modular motives [GO], and even statistical mechanics [Ba]. Many of these connections are nonobvious, but have emerged naturally as the theory of partitions has advanced. One of the questions that led to the discovery of some of these connections is, “When are the values of partition functions divisible by various integers M?”
Edgebandwidth of Tensor Product of paths and cycles
"... The bandwidth of a graph is the minimum of the maximum difference between labels of adjacent vertices in the graph. If we label the edges instead of the vertices of the graph, we can define the edgebandwidth accordingly. People start working on the edgebandwidth of graphs since 1999[7]. The edgeb ..."
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The bandwidth of a graph is the minimum of the maximum difference between labels of adjacent vertices in the graph. If we label the edges instead of the vertices of the graph, we can define the edgebandwidth accordingly. People start working on the edgebandwidth of graphs since 1999[7]. The edgebandwidth of a graph is the minimum of the maximum difference between labels of adjacent edges in the graph. Since the edgebandwidth of a graph G is equal to the bandwidth of the line graph of G, establishing the edgebandwidth of a graph is equivalent to verifying the bandwidth of one or more graphs. The decision problem corresponding to find the bandwidth of an arbitrary graph is NPcomplete[10]. It is NPcomplete even for trees of maximum degree 3[6]. Although the edgebandwidth problem is included in the bandwidth problem, the computing complexity of the edgebandwidth is unknown up to now. The application about the edgebandwidth is in the area of network circuit online routing and admission control problem. The edgebandwidth 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 edgebandwidth of the tensor product of a path with a path and a path with a cycle. Optimal edgenumberings to achieve each of these edgebandwidths are provided.