8 research outputs found

    Graph decompositions

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    This master’s thesis in graph theory focuses on graph decomposition. In practice, various forms of graph decomposition often appear in problems of arranging objects in groups with special characteristics, which is why we come across them in various fields, ranging from computer algorithms to social networks and mathematical puzzles. Graph decomposition is a union of subgraphs such that every edge of the graph belongs to exactly one subgraph. In the thesis, we examine various types of decompositions. If the subgraphs are isomorphic, we talk about an isomorphic graph decomposition. A special form of decomposition is factorisation. In this case, the graph is decomposed into spanning subgraphs. A special case of factorisation is k-factorisation, i.e., a decomposition of a graph into k-regular spanning subgraphs. When k=1, the graph is decomposed into 1-factors, which are also called perfect matchings. So 1-factorisation is a decomposition of a graph into the largest possible number of isomorphic factors, while 2-factorisation is a decomposition of a graph into a union of cycles. For illustration and a better understanding, we provide examples of graph decomposition and factorisation. In the thesis, we examine in particular the isomorphic factorisation of complete graphs. We give the proof of the Divisibility Theorem proven by Harary, Robinson and Wormald [8]. The Divisibility Theorem states that a complete graph of order n can be decomposed into t isomorphic factors if and only if t divides the number of edges of the complete graph. The proof of the theorem is constructive and includes a description of the construction of the appropriate factors for different values of parameters n and t. In the chapter on tridivisibility, we illustrate the use of this theorem by defining all nine tridivisions of the complete graph K6 and describing how we would find all 41 tridivisions of the complete graph K7. Further below we show some theorems about isomorphic decomposition of complete graphs into the greatest possible number of factors (1-factorisation), two isomorphic factors (self-complementary graphs), paths and cycles (2-factorisation). In the case of cycle decomposition, a complete graph can be decomposed into Hamiltonian cycles, cycles of arbitrary length and the shortest 3-cycles. The problem of decomposing complete graphs into 3-cycles is connected to the Steiner triple systems, so to the question of how many triplets we can make with given elements so that any pair of elements is contained in exactly one of the triplets. The decomposition of a graph into cycles of arbitrary length is at the centre of Alspach's conjecture, which was recently confirmed by D. Bryant et al. [5]. In the research of graph decomposition, there are still many unsolved conjectures. One of them is mentioned in the thesis and it refers to the decomposition of complete graphs into isomorphic subgraphs of given forms, for example, trees. Ringel and Kotzig have found that a complete graph of order 2n+1 can be decomposed into subgraphs of size n if the subgraph has a special characteristic that we today call ``gracefulness''. The conjecture is that all trees are graceful. In the end there are shown some combinatorical puzzles, which we can use at math class

    On Hamilton decompositions of infinite circulant graphs

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    The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}

    Symmetry and optimality of disjoint Hamilton cycles in star graphs

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    Multiple edge-disjoint Hamilton cycles have been obtained in labelled star graphs Stn of degree n-1, using number-theoretic means, as images of a known base 2-labelled Hamilton cycle under label-mapping auto- morphisms of Stn. However, no optimum bounds for producing such edge-disjoint Hamilton cycles have been given, and no positive or nega- tive results exist on whether Hamilton decompositions can be produced by such constructions other than a positive result for St5. We show that for all even n there exist such collections, here called symmetric collec- tions, of φ(n)/2 edge-disjoint Hamilton cycles, where φ is Euler's totient function, and that this bound cannot be improved for any even or odd n. Thus, Stn is not symmetrically Hamilton decomposable if n is not prime. Our method improves on the known bounds for numbers of any kind of edge-disjoint Hamilton cycles in star graphs

    Efficient algorithms on trees.

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    Yang, Lin.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 57-61).Abstract also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Problems and Main Results --- p.2Chapter 1.1.1 --- Firefighting on Trees --- p.2Chapter 1.1.2 --- Maximum k-Vertex Cover on Trees --- p.3Chapter 1.2 --- Background --- p.3Chapter 1.2.1 --- Random Separation --- p.4Chapter 1.2.2 --- Kernelization --- p.5Chapter 1.2.3 --- Infeasibility of Polynomial Kernel --- p.6Chapter 1.3 --- Organization of the Thesis --- p.7Chapter 2 --- Firefighting on Trees --- p.9Chapter 2.1 --- Definitions and Notation --- p.10Chapter 2.2 --- FPT Algorithms --- p.13Chapter 2.2.1 --- Saving k Vertices --- p.14Chapter 2.2.2 --- Saving k Leaves --- p.19Chapter 2.2.3 --- Protecting k Vertices --- p.23Chapter 2.3 --- Approximation --- p.29Chapter 2.3.1 --- A (1 ´ؤ 1/e)-Approximation Algorithm --- p.29Chapter 2.3.2 --- LP-Repsecting Rounding cannot Do Better --- p.33Chapter 3 --- Maximum k-Vertex Cover on Trees --- p.38Chapter 3.1 --- Maximum k Vertex Cover on Trees --- p.39Chapter 3.2 --- k-MVC on Degree Bounded Graphs --- p.45Chapter 3.3 --- k-MVC on Degeneracy Bounded Graphs --- p.46Chapter 3.4 --- Extension to Maximum k Dominating Set --- p.47Chapter 4 --- Conclusion --- p.49Chapter 4.1 --- The Firefighter problem --- p.49Chapter 4.2 --- The Maximum k-Vertex Cover problem --- p.53Acknowledgement --- p.55Bibliography --- p.5

    On Alspach's conjecture with two even cycle lengths

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    For m, n even and n > m, the obvious necessary conditions for the existence of a decomposition of the complete graph K-v when v is odd (or the complete graph with a 1-factor removed K-v\F when v is even) into v in-cycles and s n-cycles are shown to be sufficient if and only if they are sufficient for v < 7n. This result is used to settle all remaining cases with m, n less than or equal to 10. (C) 2000 Elsevier Science B.V. All rights reserved

    [[alternative]]Research on the Decompositions of Complete Bipartite Graph

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    計畫編號:NSC88-2115-M032-003研究期間:199808~199907研究經費:310,000[[sponsorship]]行政院國家科學委員

    Research problems

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