A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas
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On Excluded Minors and Biased Graph Representations of Frame Matroids
On excluded minors and biased graph representations of frame matroids by Daryl Funk M.Sc., University of Victoria, 2009 B.Sc., Simon Fraser University, 1992 Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics Faculty of Science c Daryl Funk 2015 SIMON FRASER UNIVERSITY Spring 2015 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for \Fair Dealing." Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Daryl Funk Degree: Doctor of Philosophy (Mathematics) Title of Thesis: On excluded minors and biased graph representations of frame matroids Examining Committee: Dr. Jonathan Jedwab, Chair Professor, Department of Mathematics Dr. Matthew DeVos Senior Supervisor Associate Professor, Department of Mathematics Dr. Luis Goddyn Co-Supervisor Professor, Department of Mathematics Dr. Bojan Mohar Internal Examiner Professor, Department of Mathematics Dr. Daniel Slilaty External Examiner Professor, Department of Department of Mathematics and Statistics Wright State University Date Defended: 8 January 2015 ii Partial Copyright Licence iii ABSTRACT A biased graph is a graph in which every cycle has been given a bias, either balanced or un- balanced. Biased graphs provide representations for an important class of matroids, the frame matroids. As with graphs, we may take minors of biased graphs and of matroids, and a family of biased graphs or matroids is minor-closed if it contains every minor of every member of the family. -
On Pancyclism in Hamiltonian Graphs
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Mathematics 251 (2002) 119–127 www.elsevier.com/locate/disc On pancyclism in hamiltonian graphs Mekkia Kouidera, Antoni Marczykb; ∗ aL.R.I, Universiteà de Paris-Sud, Bat.ˆ 490, 91405 Orsay, France bFaculty of Applied Mathematics, AGH, Al. Mickiewicza 30, 30-059 Krakow,à Poland Received 11 May 1999; revised 5 November 1999; accepted 3 June 2001 Abstract We investigate the set of cycle lengths occurring in a hamiltonian graph with at least one or two vertices of large degree. We prove that in every case this set contains all the integers between 3 and some t, where t depends on the order of the graph and the degrees of vertices. c 2002 Elsevier Science B.V. All rights reserved. MSC: 05C38; 05C45 Keywords: Hamiltonian graphs; Pancyclic graphs; Cycles 1. Introduction Throughout this paper, we consider only ÿnite, undirected and simple graphs. For a graph G we denote by V = V (G) its vertex set, by E = E(G) its set of edges. By Cp we mean a p-cycle of G, i.e., a cycle of length p. The vertices of a graph G of order n will be denoted by integers 0; 1; 2;:::;n− 1 and considered modulo n.We write EG(k)=E(k) for the set of edges that are incident to a vertex k. For a subgraph H of G and a vertex k of G; dH (k) denotes the number of all edges (k; j)ofE(k) with j ∈ V (H), i.e., if k ∈ V (H) then dH (k) is the degree of k in the subgraph of G induced by the set V (H). -
On the Pancyclicity of 1-Tough Graphs 1
Discrete Mathematics Letters Discrete Math. Lett. 5 (2021) 1–4 www.dmlett.com DOI: 10.47443/dml.2020.0046 Research Article On the pancyclicity of 1-tough graphs Rao Li∗ Department of Mathematical Sciences, University of South Carolina Aiken, Aiken, SC 29801, USA (Received: 14 August 2020. Received in revised form: 5 October 2020. Accepted: 13 November 2020. Published online: 23 November 2020.) c 2020 the author. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/) Abstract Let G be a graph of order n ≥ 6r−2, size e, and minimum degree δ ≥ r, where r is an integer greater than 1. The main result obtained in this note is that if G is 1-tough with the degree sequence (d1; d2; ··· ; dn) and if e ≥ ((n−r)(n−r−1)+r(2r−1))=2, then G is pancyclic, Hamiltonian bipartite, or Hamiltonian such that d1 = d2 = ··· = dk = k, dk+1 = dk+2 = ··· = dn−k+1 = n − k − 1, and dn−k+2 = dn−k+3 = ··· = dn = n − 1, where k < n=2. This result implies that the following conjecture of Hoa, posed in 2002, is true under the conditions n ≥ 40 and δ ≥ 7: every path-tough graph on n vertices and with at least ((n − 6)(n − 7) + 34)=2 edges is Hamiltonian. Using the main result of this note, additional sufficient conditions for 1-tough graphs to be pancyclic are also obtained. Keywords: pancyclic graph; 1-tough graph; Hamiltonian graph. 2020 Mathematics Subject Classification: 05C45. -
A Neighborhood and Degree Condition for Pancyclicity and Vertex Pancyclicity
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 40 (2008), Pages 15–25 A neighborhood and degree condition for pancyclicity and vertex pancyclicity Ryota Matsubara Masao Tsugaki Department of Mathematical Information Science Tokyo University of Science 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 Japan [email protected] [email protected] Tomoki Yamashita Department of Mathematics School of Dentistry, Asahi University 1851 Hozumi, Gifu 501–0296 Japan [email protected] Abstract Let G be a 2-connected graph of order n. For any u ∈ V (G) and l ∈ {m, m + 1,...,n}, if G has a cycle of length l, then G is called [m, n]- pancyclic, and if G has a cycle of length l which contains u, then G is called [m, n]-vertex pancyclic. Let δ(G) be a minimum degree of G and let NG(x) be the neighborhood of a vertex x in G. In [Australas. J. Combin. 12 (1995), 81–91] Liu, Lou and Zhao proved that if |NG(u) ∪ NG(v)| + δ(G) ≥ n+1 for any nonadjacent vertices u, v of G, then G is [3, n]-vertex pancyclic. In this paper, we prove if n ≥ 6 and |NG(u)∪NG(v)|+dG(w) ≥ n for every triple independent vertices u,v,w of G, then (i) G is [3, n]- pancyclic or isomorphic to the complete bipartite graph Kn/2,n/2, and (ii) G is [5, n]-vertex pancyclic or isomorphic to the complete bipartite graph Kn/2,n/2. 1 Introduction In this paper, we consider only finite graphs without loops or multiple edges. -
Vertex Pancyclic Graphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 120 (2002) 219–237 Vertex pancyclic graphs Bert Randeratha; ∗,Ingo Schiermeyer b,Meike Tewes b,Lutz Volkmann c aInstitut fur Informatik, Universitat zu Koln, Pohligstrasse 1 50969 Koln, Germany bFakultat fur Mathematik und Informatik, TU Bergakademie Freiberg, 09596 Freiberg, Germany cLehrstuhl II fur Mathematik, RWTH Aachen, 52056 Aachen, Germany Received 17 September 1999; received in revised form 31 July 2000; accepted 21 July 2001 Abstract Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3 6 k 6 n,and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3 6 k 6 n. In this paper,we shall present di0erent su1cient conditions for graphs to be vertex pancyclic. ? 2002 Elsevier Science B.V. All rights reserved. 1. Terminology We consider ÿnite,undirected,and simple graphs G with the vertex set V (G) and the edge set E(G). The order of a graph G is denoted by n and the size by m, respectively. The complete graph of order n is denoted by Kn,and the complete bipartite graph with the partite sets A and B with |A| = p and |B| = q is denoted by Kp;q. A graph G is k-connected if G − S is connected for every subset S ⊆ V (G) with |S| 6 k − 1. A vertex v of a connected graph G is called a cut vertex if G − v is disconnected. -
Graph Theory Graph Theory (II)
J.A. Bondy U.S.R. Murty Graph Theory (II) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. -
ON K-PATH PANCYCLIC GRAPHS
Discussiones Mathematicae Graph Theory 35 (2015) 271–281 doi:10.7151/dmgt.1795 ON k-PATH PANCYCLIC GRAPHS Zhenming Bi and Ping Zhang Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA e-mail: [email protected] Abstract For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k +1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G is k-path pancyclic Keywords: Hamiltonian, panconnected, pancyclic, path Hamiltonian, path pancyclic. 2010 Mathematics Subject Classification: 05C45, 05C38. 1. Introduction A Hamiltonian cycle in a graph G is a cycle containing every vertex of G and a graph having a Hamiltonian cycle is a Hamiltonian graph. The first theoretical result on Hamiltonian graphs occurred in 1952 and is due to Dirac [7]. Theorem 1.1 (Dirac). If G is a graph of order n ≥ 3 such that the minimum degree δ(G) ≥ n/2, then G is Hamiltonian. For a nontrivial graph G that is not complete, define σ2(G) = min{deg u + deg v : uv∈ / E(G)} where deg w is the degree of a vertex w in G. -
Discrete Mathematics Forbidden Subgraphs for Chorded Pancyclicity
Discrete Mathematics 340 (2017) 2878–2888 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Forbidden subgraphs for chorded pancyclicity Megan Cream a, Ronald J. Gould b, Victor Larsen c,* a Department of Mathematics, Spelman College, 350 Spelman Lane SW, Atlanta, GA 30314, United States b Department of Mathematics and Computer Science, Emory University, 400 Dowman Drive, Atlanta, GA 30322, United States c Department of Mathematics, Kennesaw State University, 1100 S. Marietta Parkway, Marietta, GA 30060, United States article info a b s t r a c t Article history: We call a graph G pancyclic if it contains at least one cycle of every possible length m, for Received 26 October 2016 3 ≤ m ≤ jV (G)j. In this paper, we define a new property called chorded pancyclicity. We Received in revised form 26 July 2017 explore forbidden subgraphs in claw-free graphs sufficient to imply that the graph contains Accepted 30 July 2017 at least one chorded cycle of every possible length 4; 5;:::; jV (G)j. In particular, certain Available online 1 September 2017 paths and triangles with pendant paths are forbidden. ' 2017 Elsevier B.V. All rights reserved. Keywords: Pancyclic Chorded cycle Forbidden subgraph Hamiltonian 1. Introduction In the past, forbidden subgraphs for Hamiltonian properties in graphs have been widely studied (for an overview, see [1]). A graph containing a cycle of every possible length from three to the order of the graph is called pancyclic. The property of pancyclicity is well-studied. In this paper, we define the notion of chorded pancyclicity, and study forbidden subgraph results for chorded pancyclicity. -
Results on Pancyclc Graphs and Complete Hamiltonian Graphs
International Journal of Engineering Applied Sciences and Technology, 2017 Vol. 2, Issue 5, ISSN No. 2455-2143, Pages 33-38 Published Online March-April 2017 in IJEAST (http://www.ijeast.com) RESULTS ON PANCYCLC GRAPHS AND COMPLETE HAMILTONIAN GRAPHS S.VENU MADHAVA SARMA T.V.PRADEEP KUMAR Research Scholar, Assistant Professor of Mathematics Department of Mathematics, ANU College of Engineering and Technology, Rayalaseema University, Acharya Nagarjuna University Kurnool, Andhra Pradesh. Guntur INDIA Abstract - In 1856, Hamiltonian introduced the In 1913, H. Dudeney mentioned a puzzle problem. Hamiltonian Graph where a Graph which is Eventhough the four color problem was invented it covered all the vertices without repetition and end was solved only after a century by Kenneth Appel with starting vertex. In this paper I would like to and Wolfgang Haken. prove that Relation between Complete Hamiltonian Graph and Pan-Cyclic Graphs This time is considered as the birth of Graph Theory. Key Words: Graph, Complete Graph, Bipartite Cayley studied particular analytical forms from Graph Hamiltonian Graph and Pan-Cyclic Graphs. differential calculus to study the trees. This had many implications in theoretical chemistry. This lead to the I. INTRODUCTION invention of enumerative graph theory. The origin of graph theory started with the problem Any how the term “Graph” was introduced by of Koinsber bridge, in 1735. Sylvester in 1878 where he drew an analogy between “Quantic invariants” and covariants of algebra and This problem lead to the concept of Eulerian Graph. molecular diagrams. Euler studied the problem of Koinsberg bridge and constructed a structure to solve the problem called In 1941, Ramsey worked on colorations which lead Eulerian graph. -
Graph Theory Graph Theory (III)
J.A. Bondy U.S.R. Murty Graph Theory (III) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. -
Arxiv:1708.08132V1 [Math.CO] 27 Aug 2017 Ftevsiivko Nain Hoy[3.Te R Eyueu for Useful Very 26]
TOPOLOGICAL TUTTE POLYNOMIAL SERGEI CHMUTOV Abstract. This is a survey of recent works on topological extensions of the Tutte polynomial. Mathematics Subject Classifications: 05C31, 05C22, 05C10, 57M15, 57M25, 57M30 1. Introduction We survey recent works on topological extensions of the Tutte polynomial. The extensions deal with graphs on surfaces. Graphs embedded into surfaces are the main objects of the topological graph theory. There are several books devoted to this subject [23, 28, 38, 43]. Cellular embeddings of graphs can be described as ribbon graphs. Oriented ribbon graphs appear under different names such as rotation systems [43], maps, fat graphs, cyclic graphs, dessins d’enfants [38]. Since the pioneering paper of L. Heffter in 1891 [31] they occur in various parts of mathematics ranging from graph theory, combinatorics, and topology to representation theory, Galois theory, algebraic geometry, and quantum field theory [13, 30, 35, 38, 45, 46]. For example, ribbon graphs are used to enumerate cells in the cell decomposition of the moduli spaces of complex algebraic curves [30, 45, 38]. The absolute Galois group Aut(Q/Q) faithfully acts on the set of ribbon graphs (see [38] and references therein). Ribbon graphs are the main combinatorial objects of the Vassiliev knot invariant theory [13]. They are very useful for Hamiltonicity of the Cayley graphs [25, 26]. M. Las Vergnas [40] found a generalization of the Tutte polynomial to cellu- larly embedded graphs as an application of his matroid perspectives. Recently his polynomial was extended to not necessarily cellularly embedded graphs [24]. In 2001 B. Bollob´as and O. Riordan [4, 5], motivated by some problems of knot theory, introduced a different generalization of the Tutte polynomial for ribbon graphs. -
Knot Theory: History and Applications with a Connection to Graph Theory
KNOT THEORY: HISTORY AND APPLICATIONS WITH A CONNECTION TO GRAPH THEORY A Thesis Submitted to the Faculty of the North Dakota State University of Agriculture and Applied Science By Mitchel Todd Keller In Partial Fulfillment of the Requirements for the Degree of BACHELOR OF SCIENCE and the UNIVERSITY HONORS PROGRAM Major Department: Mathematics April 2004 Fargo, North Dakota ABSTRACT Keller, Mitchel Todd, B.S., Department of Mathematics, College of Science and Math- ematics, North Dakota State University, April 2004. Knot Theory: History and Ap- plications with a Connection to Graph Theory. Major Professor: Dr. Jorge Alberto Calvo. This honors thesis introduces some fundamental ideas of knot theory in a way that is accessible to nonmathematicians. It summarizes some of the major histori- cal developments in the mathematical theory of knots, beginning with Thomson and Tait and ending with some of the important results of the late 20th century. A few important examples of ways in which knot theory can be used to model real-world phenomena are discussed, including the importance of topology to the pharmaceu- tical industry. It concludes by using chain complexes of based, finitely-generated Z-modules to study the Laplacians of signed plane graphs and to extend a theorem of Lien and Watkins [14] regarding the Goeritz equivalence of the signed Laplacians of a signed plane graph and its dual by showing that it is possible to use only ( 1)- ± diagonal forms instead of the (0, 1)-diagonal forms used by Lien and Watkins. ± ii ACKNOWLEDGMENTS I would like to thank Professor J.A. Calvo for serving as my advisor for this thesis and the majority of my undergraduate career.