1. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada. The first author thanks the joint CNRS/INRIA/UNSA project SLOOP for its hospitality while this paper was being prepared. Cycle Decompositions of Kn and Kn−I. By Brian Alspach and Heather Gavlas. Get PDF 188 KB Cite. BibTex; Full citation; Abstract. AbstractWe establish necessary and sufficient conditions for decomposing the complete graph of even order minus a 1-factor into even cycles and the complete graph of odd order into odd cycles. It is conjectured that if the minimum number of odd cycles in a cycle decomposition of an Eulerian graph G with m edges is a and the maximum number of odd cycles in a cycle decomposition is c. the cycle length m divides the number of edges in either K,, that is, 9, nn 1 or Kn - I, that is, 9. B. Alspach and H. Gavlas [l] have shown that for the case when m and n are either both odd or both even, the necessary conditions are also sufficient. In this thesis we extend their results to the case m even, n odd, and m odd, n even. That.
the rst Hamilton cycle until returning to the central vertex. Then move into the next Hamilton cycle in clockwise order and continue in this way until completing the tour. This Euler tour of K n, nodd, has the very nice property that any n 3 or fewer successive edges form a path. Thus, if ‘ n 3, we can chop the tour up into paths of length ‘. Let Ck denote a cycle of length k and let Sk denote a star with k edges. For multigraphs F, G and H, an F, G-decomposition of H is an edge decomposition of H into copies of F and G using at.
Alspach's conjecture, posed by Alspach in 1981, concerns the characterization of disjoint cycle covers of complete graphs with prescribed cycle lengths. With Heather Gavlas Jordon, in 2001, Alspach proved a special case, on the decomposition of complete graphs into cycles. Ck denotes a cycle of length k. Ck −factor is a spanning subgraph H of G such that each component of H is a Ck. Partitioning the edge set of G into Ck −factors is called a Ck −factorization. Cˆ k is a cycle of length k of a m−partite graph having vertices in all the partite sets where k ≥ m. The Oberwolfach Problem asks whether it is possible to decompose the complete graph on 2nl vertices or the complete graph on 2n vertices with a spanning set of independent edges removed into isomorphic factors each comprising a set of cycles whose combined length is 2nl or 2n, respectively. We trace the. graph Kn can be deccmposed into C 's and one K where 3 5 Cr denotes a cycle of length r. This together with the fact that a Steiner triple system is known to exist for n Z 1 or 3 mod 6 e&ablishes -, the exiftence of a pairwise balanced design n; 5,3;1 for any odd d' In Chapter.3 it is shdwn that the necessary condit'ions. In the ten years since the publication of the best-selling first edition, more than 1,000 graph theory papers have been published each year. Reflecting these advances, Handbook of Graph Theory, Second Edition provides comprehensive coverage of the main topics in pure and applied graph theory.
Cycle decompositions. III. Complete graphs and fixed length cycles. J. Combin. Des. 10 2002, no. 1, 27–78. On decomposing Kn−I into cycles of a fixed odd length. Algebraic and topological methods in graph theory Lake Bled, 1999. Discrete Math. pdf. Self-complementary graphs and generalisations: A comprehensive reference manual. University of Malta, 1999. Alastair Farrugia. Download with Google Download with Facebook or download with email. Self-complementary graphs and generalisations: A comprehensive reference manual. Download. JOURNAL OF COMBINATORIAL THEORY B 20, 222-225 1976 Path Numbers of Tournaments BRIAN ALSPACH Simon Fraser University, Burnaby V5A I S6, Canada DAVID W. MASON Directorate of Logistics Analysis, National Defence Headquarters, Ottawa, Canada AND NORMAN J. PULLMAN Jeffery Hall, Queen's University, Kingston K7L 3N6, Canada Communicated by P. Erdds Received.
Brian Alspach, Danny Dyer, Denis Hanson, Boting Yang, Lower bounds on edge searching, Proceedings of the First international conference on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies, p.516-527, April 07-09, 2007, Hangzhou, China. Download as PDF, TXT or read online from Scribd. Flag for inappropriate content. Download. The problem has been solved completely by Brian Alspach et al. ;. Alspach and H. Gavlas, Cycle decompositions of Kn and Kn I, J. Combin. Theory Ser.B 81 2001,77-99.
To create a de Bruijn sequence of order n, splice one cycle into the other by writing the first cycle up to and including a common n − 1-tuple, then writing the second cycle starting from the common n − 1-tuple and cycling around to write the full cycle, finishing again with the common n − 1-tuple, and finally completing the first. Motivated by work in graph theory, we define the fixing number for a matroid. We give upper and lower bounds for fixing numbers for a general matroid in terms of the size and maximum orbit size under the action of the matroid automorphism group.
mit Heather Gavlas: Cycle Decompositions of Kn and Kn−I. Journal of Combinatorial Theory, Reihe B, Band 81, 2001, S. 77–99 Journal of Combinatorial Theory, Reihe B, Band 81, 2001, S. 77–99 mit Marni Mishna: Enumeration of Cayley graphs and digraphs, Discrete Math., Band 256, 2002, S. 527–539. Readbag users suggest that sc-graph-survey.pdf is worth reading. The file contains 270 pages and is free to view, download or print.
 Darryn Bryant and Peter Danziger, On bipartite 2-factorisations of Kn - I and the Oberwolfach problem, Journal of Graph Theory, 68 2011 22--37.  Darryn Bryant, Daniel Horsley, Barbara Maenhaut and Benjamin R. Smith, Cycle decompositions of complete multigraphs, Journal of Combinatorial Designs, 19 2011 42--69. this PDF file - The Electronic Journal of Combinatorics advertisement A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas Compiled by Thomas Zaslavsky Department of Mathematical Sciences Binghamton University SUNY Binghamton, New. A cycle is even if its length is even; otherwise it is odd. A cycle of length n is an n-cycle; a 3-cycle is also called a triangle. A graph of order n that is a path or a cycle is denoted by P n or C n, respectively. We now consider a very basic concept in graph theory, namely COnnected and disconnected graphs.
Cycle decomposition of and Edit.Brian Alspach and Heather Gavlas established necessary and sufficient conditions for the existence of a decomposition of a complete graph of even order minus a 1-factor into even cycles and a complete graph of odd order into odd cycles. Their proof relies on Cayley graphs, in particular, circulant graphs. Her research interests are algebraic combinatorics and colorings, distance and convexity, traversability, decompositions, and domination within graph theory. xvi CONTRIBUTORS Alfred V. Aho Columbia University Maria Chudnovsky Columbia University Brian Alspach University of Newcastle, Australia Fan Chung University of California, San Diego.
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