Hamilton cycles in strong products of graphs

“…In this section, we just introduce less standard concepts which we use and we also recall some results on factors in strong products of graphs mostly from [10,11].…”

Section: Preliminariesmentioning

confidence: 99%

Hamiltonian threshold for strong products of graphs

Kráľ

Stacho

2008

Journal of Graph Theory

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“…Pilśniak in [11] showed that the distinguishing index of traceable graphs, graphs with a Hamiltonian path, of order equal or greater than seven is at most two. Also Král et al in [9] showed that if G 0 , . .…”

Section: Distinguishing Index Of Strong Product Of Two Graphsmentioning

confidence: 99%

Distinguishing number and distinguishing index of strong product of two graphs

Soltani,

Alikhani

2017

Preprint

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The distinguishing number (index) D(G) (D ′ (G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. The strong product G ⊠ H of two graphs G and H is the graph with vertex set V (G) × V (H) and edge set {{(x 1 , x 2 ), (y 1 , y 2 )}|x i y i ∈ E(G i ) or x i = y i for each 1 ≤ i ≤ 2.}. In this paper we study the distinguishing number and the distinguishing index of strong product of two graphs. We prove that for every k ≥ 2, the k-th strong power of a connected S-thin graph G has distinguishing index equal 2.

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“…For example, the central problem of determining whether there is a Hamiltonian cycle in a given graph is NP-complete. Therefore, many research works focus on the existence of Hamiltonian cycles in some special classes of graphs and polynomial algorithms to find them, such as [53], [15] and [70]. On the other hand, sufficient conditions for Hamiltonicity of graphs have taken up a great proportion of the theoretical results.…”

Section: Introductionmentioning

confidence: 99%

Paths, Cycles and Related Partitioning Problems in Graphs

Zhang¹

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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...

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Hamilton cycles in strong products of graphs

Abstract: We prove that the strong product of any n connected graphs of maximum degree at most n contains a Hamilton cycle. In particular, Gis hamiltonian for each connected graph G, which answers in affirmative a conjecture of Bermond, Germa, and Heydemann. ------------------

Cited by 7 publications

References 9 publications

Hamiltonian threshold for strong products of graphs

Hamiltonian threshold for strong products of graphs

Distinguishing number and distinguishing index of strong product of two graphs

Paths, Cycles and Related Partitioning Problems in Graphs

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