Hamiltonian Spider Intersection Graphs Are Cycle Extendable

Abueida, Atif A.; Busch, Arthur H.; Sritharan, R.

doi:10.1137/130914164

Cited by 9 publications

(18 citation statements)

References 19 publications

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“…A perfect elimination ordering of a graph G is an ordering, v 1 A set of vertices X ⊆ V (G) which induces a complete subgraph of G is a clique.…”

Section: Counterexamples To Hendry's Conjecturementioning

confidence: 99%

Hamiltonian Chordal Graphs are not Cycle Extendable

Lafond¹,

Seamone²

2015

SIAM J. Discrete Math.

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In 1990, Hendry Conjectured that every Hamiltonian chordal graph is cycle extendable; that is, the vertices of any non-Hamiltonian cycle are contained in a cycle of length one greater. We disprove this conjecture by constructing counterexamples on n vertices for any n ≥ 15. Furthermore, we show that there exist counterexamples where the ratio of the length of a nonextendable cycle to the total number of vertices can be made arbitrarily small. We then consider cycle extendability in Hamiltonian chordal graphs where certain induced subgraphs are forbidden, notably Pn and the bull. Introduction.All graphs considered here are simple, finite, and undirected. A graph is Hamiltonian if it has a cycle containing all vertices; such a cycle is a Hamiltonian cycle. A graph G on n vertices is pancyclic if G contains a cycle of length m for every integer 3 ≤ m ≤ n. Let C and C be cycles in G of length m and m + 1, respectively, such that V (C ) \ V (C) = {v}. We say that C is an extension of C and that C is extendable (or, C extends through v to C ). If every non-Hamiltonian cycle of G is extendable, then G is cycle extendable. If, in addition, every vertex of G is contained in a triangle, then G is fully cycle extendable. The study of pancyclic graphs was initiated by Bondy [3], who recognized that most of the sufficient conditions for Hamiltonicity known at the time in fact implied a more complex cycle structure. Hendry [12] introduced the concept of cycle extendability and proved that many known sufficient conditions for a graph to be pancyclic in fact were sufficient for a graph to be (fully) cycle extendable.Given a graph G and a set of vertices U ⊆ V (G), we denote by G[U ] the subgraph obtained by deleting from G all vertices except those in U ; G[U ] is the subgraph induced by U , and a subgraph of G is an induced subgraph if it is induced by some U ⊆ V (G). A graph is chordal if it contains no induced cycles of length 4 or greater. It is not hard to show that every Hamiltonian chordal graph is pancyclic (see Proposition 3.4); however, the question of whether not every Hamiltonian chordal graph is cycle extendable has remained open since 1990.Conjecture 1.1 (Hendry's Conjecture [12]). If G is a Hamiltonian chordal graph, then G is fully cycle extendable.In this paper, we settle Hendry's Conjecture in the negative. In section 2, we show that (a) for any n ≥ 15 there exists a counterexample to Hendry's Conjecture on n vertices and (b) for every real number α > 0 there exists a counterexample G with a nonextendable cycle C such that |V (C)| < α|V (G)|. The question then remains: *

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“…A perfect elimination ordering of a graph G is an ordering, v 1 A set of vertices X ⊆ V (G) which induces a complete subgraph of G is a clique.…”

Section: Counterexamples To Hendry's Conjecturementioning

confidence: 99%

Hamiltonian Chordal Graphs are not Cycle Extendable

Lafond¹,

Seamone²

2015

SIAM J. Discrete Math.

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“…This problem can be understood as a query on the equivalence between Hamiltonicity and cycle extendability in chordal graphs. For several subclasses of chordal graphs, this equivalent relation has been verified ( [33], [3] and [2]). However, the equivalence is not valid in general chordal graphs, as has been shown in [76] recently.…”

Section: Introductionmentioning

confidence: 76%

“…If s i0 forms a vertex-cycle, then by Property 2 and Property 3, the path P (s i1 , s i(n+1) ) must form a multicolored cycle with some vertices from ∪ n i=1 U i in P . Furthermore, by Property 4 and Property 6, P (s i1 , s i(n+1) ) must form a multicolored cycle with some paths P (u 1 j , u 2 j ), where u j ∈ s i for some 1 ≤ j ≤ n. All such cycles belong to cycle kind of (2). Finally, every cycle that contains only vertices from ∪ n i=1 U i is of kind (4), by Property 5.…”

Section: = C ∪ {S} E(g) = E(g) \ E(s)mentioning

confidence: 99%

“…Therefore, according to the direction of the arcs between V (P ) and v, we can classify v into three categories, as follows. We call a vertex v ∈ V (D)\V (P ) a dominating, dominated or hybrid vertex of P if v belongs to category (1), (2) or (3), respectively. Furthermore, for a hybrid vertex v in category (3) we say that it switches at s. Note that the definitions are applicable to all paths (not necessarily non-extendable paths).…”

Section: Conjecture 52 a Tournament T Satisfying P Ath αNmentioning

confidence: 99%

“…It remained open for nearly three decades. Although several supporting results appeared (Abueida et al [2], Abueida and Sritharan [3], and Chen et al [33]), showing that the answer is yes when restricted to some special classes of graphs, the general problem was answered negatively recently by counterexamples constructed by Lafond and Seamone ( [76], 2013). For digraphs, related research has been focused mainly on cycle extendability in tournaments and in-tournaments (Tewes and Volkmann [100], and Meierling [87]).…”

Section: Introduction Terminology and Notationmentioning

confidence: 99%

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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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Toughness and Hamiltonicity of strictly chordal graphs

Markenzon

Waga

2016

Int Trans Operational Res

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The clique‐based structure of chordal graphs allows the development of efficient solutions for many algorithmic problems. In this context, the minimal vertex separators play a decisive role. In this paper, we study properties of these structures, presenting a linear time determination of the toughness of strictly chordal graphs. We prove that every 1‐tough strictly chordal graph is Hamiltonian. This result leads to the characterization of a new class, the Hamiltonian strictly chordal graphs. We also prove that graphs belonging to this class are cycle extendable.

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Hamiltonian Spider Intersection Graphs Are Cycle Extendable

Cited by 9 publications

References 19 publications

Hamiltonian Chordal Graphs are not Cycle Extendable

Hamiltonian Chordal Graphs are not Cycle Extendable

Paths, Cycles and Related Partitioning Problems in Graphs

Toughness and Hamiltonicity of strictly chordal graphs

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