SPRINGER BRIEFS IN MATHEMATICS John C. George Abdollah Khodkar W.D. Wallis Pancyclic and Bipancyclic Graphs 123 SpringerBriefs in Mathematics Series Editors Nicola Bellomo Michele Benzi Palle Jorgensen Tatsien Li Roderick Melnik Otmar Scherzer Benjamin Steinberg Lothar Reichel Yuri Tschinkel G. George Yin Ping Zhang SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. More information about this series at http://www.springer.com/series/10030 John C. George • Abdollah Khodkar • W.D. Wallis Pancyclic and Bipancyclic Graphs 123 John C. George Abdollah Khodkar Department of Mathematics Department of Mathematics and Computer Science University of West Georgia Gordon State College Carrollton, GA, USA Barnesville, GA, USA W.D. Wallis Department of Mathematics Southern Illinois University Evansville, IN, USA ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-319-31950-6 ISBN 978-3-319-31951-3 (eBook) DOI 10.1007/978-3-319-31951-3 Library of Congress Control Number: 2016935702 © The Author(s) 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland The authors would like to dedicate this book to our families: Amanda and Robert (JCG); Sarah, Arvin, and Darian (AK); Ann (WDW) Preface For nearly 50 years, there has been some interest in the cycles occurring as subgraphs of graphs. In 1971, Adrian Bondy introduced the idea of a pancyclic graph, one that contained cycles of every possible length. But from that time, the idea has been largely unexplored. Together with some of our colleagues, we have been looking at pancyclicity, and we decided it was time to write a book on the subject and related areas. It is our hope that students and researchers alike will find in this volume inspiration and ideas to facilitate their own work and shed further light on this fascinating topic. We would like to express our thanks to some colleagues who have helped us in our endeavors, including Saad El-Zanati, Alison Marr, Alex Peterson, Nick Phillips, Christina Wahl, and Zach Walsh. Our first draft was met with some very useful report from the referees, which we used to improve the book. We wish to thank them for their input. We also wish to express our thanks to the staff of Springer and in particular to Razia Amzad for her help. Detroit, MI, USA John C. George Carrollton, GA, USA Abdollah Khodkar Evansville, IN, USA W.D. Wallis vii Contents 1 Graphs ........................................................................ 1 1.1 Introduction ............................................................. 1 1.2 Graphs: The Basics ..................................................... 1 1.3 Products ................................................................. 3 1.4 Walks, Paths, and Cycles ............................................... 4 1.5 Colorings and Cycles ................................................... 6 2 Degrees and Hamiltoneity................................................... 9 2.1 A Theorem of Chvátal .................................................. 9 2.2 A Theorem of Fan....................................................... 10 2.3 A Theorem of Bondy and Its Generalization .......................... 12 3 Pancyclicity ................................................................... 21 3.1 Introduction ............................................................. 21 3.2 Bounds................................................................... 22 3.3 Pancyclic Graph Products .............................................. 32 3.4 Open Problems .......................................................... 34 4 Minimal Pancyclicity ........................................................ 35 4.1 Introduction ............................................................. 35 4.2 Minimal Pancyclic Graphs: Small Orders ............................. 36 4.2.1 Fewer Than Two Chords ....................................... 37 4.2.2 Two Chords ..................................................... 37 4.2.3 Three Chords.................................................... 38 4.3 Four Chords ............................................................. 41 4.4 Five Chords.............................................................. 42 4.5 More General Bounds for Pancyclics .................................. 42 5 Uniquely Pancyclic Graphs ................................................. 49 5.1 Introduction ............................................................. 49 5.2 Small Cases ............................................................. 49 5.3 Outerplanar UPC Graphs ............................................... 51 5.4 More General UPC Graphs ............................................. 53 ix x Contents 5.5 Cycle Space of a Graph ................................................. 62 5.6 Bounds on the Number of Edges in a UPC Graph .................... 64 5.7 Open Problems .......................................................... 67 6 Bipancyclic Graphs .......................................................... 69 6.1 Introduction ............................................................. 69 6.2 Edge Number Conditions ............................................... 69 6.3 Degree Conditions ...................................................... 71 7 Uniquely Bipancyclic Graphs............................................... 81 7.1 Introduction ............................................................. 81 7.2 Graphs with Fewer than Two Chords .................................. 82 7.3 Two Chords.............................................................. 82 7.4 Three Chords ............................................................ 84 7.5 More Chords: Computer Searches ..................................... 96 8 Minimal Bipancyclicity ...................................................... 99 8.1 Introduction ............................................................. 99 8.2 Minimal Bipancyclic Graphs with Excess Less than 2................ 100 8.3 Excess 2 ................................................................. 100 8.4 Excess 3 ................................................................. 101 8.5 Excess 4 ................................................................. 102 8.6 More General Bounds for Bipancyclics................................ 103 8.7 Bipancyclic Graph Products ............................................ 105 References......................................................................... 107 List of Figures Fig. 1.1 Two representations of the graph K4 ................................. 3 Fig. 1.2 The graph K3;5 ........................................................ 3 Fig. 2.1 Case where k is as large as possible.................................. 11 Fig. 2.2 Case where s k C 2 ................................................ 11 Fig. 2.3 Case where s D k C 1 ................................................ 12 Fig. 2.4 P with only one point (an endpoint) on C ........................... 13 Fig. 2.5 x0 is the furthest neighbor of x........................................ 14 Fig. 2.6 r and s are on the same path .......................................... 14 Fig. 2.7 r and s are on different paths ......................................... 14 Fig. 2.8 The C-path in case 1 .................................................. 15 Fig. 2.9 The C-path in case 2 .................................................. 15 Fig. 2.10 The required new cycle ............................................... 15 Fig. 2.11 A C-path with a1 and a2 on C ........................................ 16 Fig. 2.12 Case 1: dCnfag.x/ D 0 ................................................. 17 C C` Fig. 2.13 Case 2: dCnfag.x/ 1 and a is not adjacent to b ................ 17 C C Fig. 2.14 Case 2: dCnfag.x/ 1 and a is not adjacent to v ................. 18 Fig. 2.15 k parallel edges between H and C.................................... 18 Fig. 3.1 Graph Fn .............................................................. 31 Fig. 3.2 Constructing the 2k C 1-cycle ....................................... 33 Fig. 4.1