Paths, Cycles and Related Partitioning Problems in Graphs

Abstract

P a t h s , Cy c l e s a n d Re l a t e d P a r t i t i o n i n g P r o b l e ms i n Gr a p h s P a t h s , Cy c l e s a n d Re l a t e d P a r t i t i o n i n g P r o b l e ms i n Gr a p h s Paths, Cycles and Related Partitioning Problems in Graphs Zanbo ZhangThe research was supported by and carried out in the group of Formal Methods and Tools, in the Faculty of Electrical Engineering, Mathematics and Computer Science of the University of Twente, the Netherlands. The financial support from the University of Twente for this research work and its publication is gratefully acknowledged.The research was also supported by the National Natural Science Foundation of China (NSFC, 11471003 and 11471342) Copyright c⃝2017 Zanbo Zhang, Enschede, the Netherlands. All rights reserved. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission from the copyright owner. PATHS, CYCLES AND RELATED PARTITIONING PROBLEMS IN GRAPHS DISSERTATION PrefaceThis thesis is based on joint work of the author with several different collaborators during the last five years. It is composed of a short introductory chapter, followed by five technical chapters. These five chapters are all based on associated research papers that are in different stages of submission, refereeing, acceptance or publication, and that are listed below together with several other joint publications of the author.The underlying research papers as well as the corresponding chapters in this thesis are the result of continuous part-time research efforts of the author, in collaboration with other researchers, on paths and cycles in graphs.The first chapter contains a brief introduction, with some background and motivation for the research in this field, and with some remarks on earlier work that inspired the author to contribute to this field. The details of the author's contributions are presented in the subsequent five chapters, that are all self-contained. The second chapter deals with the extremal digraphs one has to exclude when relaxing a classical degree condition for the existence of Hamiltonian cycles in digraphs. The third and fourth chapter deal with sufficient conditions for path extendability and cycle extendability in digraphs, respectively. In the fifth chapter, in the context of path and cycle properties, we study the number of 2-paths in oriented graphs and tournaments. We also present some applications to demonstrate the relevance of conditions in terms of the number of 2-paths. In the sixth chapter, we study monochromatic clique and multicolored cycle partitioning problems in edge-colored graphs, from the perspective of computational complexity and algorithmic solutions. i ii PrefaceThe thesis has been written as a collection of independent papers. To guarantee the independence and readability of the chapters, we chose to maintain the structure of journal papers, with a short introduction and background, and th...