Triple Even Star Decomposition of Complete Bipartite Graphs

Let G be a finite, connected, undirected graph without loops or multiple edges. A decomposition {G2, G4, . . . , G2k} of G is said to be an even star decomposition if each Gi  is a star and |E(Gi)| = i for all i = 2, 4, . . . , 2k.     A graph G is said to have Triple Even Star Decomposition (TESD) if G can be decomposed into 3k stars {3S2, 3S4, . . . , 3S2k}. In this paper, we characterize Triple Even Star Decomposition of complete bipartite graphs Km,n when m = 2 and m = 3

Star Decompositions of Bipartite Graphs

In Chapter 1, we will introduce the definitions and the notations used throughout this thesis. We will also survey some prior research pertaining to graph decompositions, with special emphasis on star-decompositions and decompositions of bipartite graphs. Here we will also introduce some basic algorithms and lemmas that are used in this thesis. In Chapter 2, we will focus primarily on decomposition of complete bipartite graphs. We will also cover the necessary and sufficient conditions for the decomposition of complete bipartite graphs minus a 1-factor, also known as crown graphs and show that all complete bipartite graphs and crown graphs have a decomposition into stars when certain necessary conditions for the decomposition are met. This is an extension of the results given in "On claw-decomposition of complete graphs and complete bigraphs" by Yamamoto, et. al. We will propose a construction for the decomposition of the graphs. In Chapter 3, we focus on the decomposition of complete equipartite tripartite graphs. This result is similar to the results of "On Claw-decomposition of complete multipartite graphs" by Ushio and Yamamoto. Our proof is again by construction and we propose how it might extend to equipartite multipartite graphs. We will also discuss the 3-star decomposition of complete tripartite graphs. In Chapter 4 , we will discuss the star decomposition of 4-regular bipartite graphs, with particular emphasis on the decomposition of 4-regular bipartite graphs into 3-stars. We will propose methods to extend our strategies to model the problem as an optimization problem. We will also look into the probabilistic method discussed in "Tree decomposition of Graphs" by Yuster and how we might modify the results of this paper to star decompositions of bipartite graphs. In Chapter 5, we summarize the findings in this thesis, and discuss the future work and research in star decompositions of bipartite and multipartite graphs

Decomposition of tensor product of complete graphs into cycles and stars with four edges

A. Muthusamy is partially supported by the University Grant Commission, Government of India, New Delhi and the Department of Science and Technology, New Delhi.In this paper, we prove that the necessary conditions are sufficient for the existence of a decomposition of tensor product of complete graphs into cycles and stars with four edges.Publisher's Versio

Decomposition of the complete bipartite graph with a 1-factor removed into paths and stars

Let P_k denote a path on k vertices, and let S_k denote a star with k edges. For graphs F, G, and H, a decomposition of F is a set of edge-disjoint subgraphs of F whose union is F. A (G,H)-decomposition of F is a decomposition of F into copies of G and H using at least one of each. In this paper, necessary and sufficient conditions for the existence of the (P_{k+1},S_k)-decomposition of the complete bipartite graph with a 1-factor removed are given

Oberwolfach rectangular table negotiation problem

AbstractWe completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H(k,3) be a bipartite graph with bipartition X={x1,x2,…,xk},Y={y1,y2,…,yk} and edges x1y1,x1y2,xkyk−1,xkyk, and xiyi−1,xiyi,xiyi+1 for i=2,3,…,k−1. We completely characterize all complete bipartite graphs Kn,n that can be factorized into factors isomorphic to G=mH(k,3), where k is odd and mH(k,3) is the graph consisting of m disjoint copies of H(k,3)

Fork-decomposition of strong product of graphs

Decomposition of arbitrary graphs into subgraphs of small size is assuming importance in the literature. There are several studies on the isomorphic decomposition of graphs into paths, cycles, trees, stars, sunlet etc. Fork is a tree obtained by subdividing any edge of a star of size three exactly once. In this paper, we investigate the necessary and sufficient for the fork-decomposition of Strong product of graphs