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Toughness, hamiltonicity and split graphs

Dieter Kratsch, Jenő Lehel, Haiko Müller
1996 Discrete Mathematics  
First, we consider properties of graph classes related to hamiltonicity, traceability and toughness concepts and display some algorithmic consequences.  ...  Related to Chvfital's famous conjecture stating that every 2-tough graph is hamiltonian, we study the relation of toughness and hamiltonieity on special classes of graphs.  ...  We are going to show that our three properties of graph classes are indeed strongly related to each other (without the need of any structural properties of a certain graph class). Theorem 2.2.  ... 
doi:10.1016/0012-365x(95)00190-8 fatcat:3vqxxzp2sfc5tlxhqv6xefmt3m

Graph connectivity and binomial edge ideals

Arindam Banerjee, Luis Núñez-Betancourt
2016 Proceedings of the American Mathematical Society  
We relate homological properties of a binomial edge ideal J G to invariants that measure the connectivity of a simple graph G.  ...  Specifically, we show if R/J G is a Cohen-Macaulay ring, then graph toughness of G is exactly 1 2 . We also give an inequality between the depth of R/J G and the vertexconnectivity of G.  ...  Graph toughness and binomial edge ideals. In this subsection we establish relations between graph toughness and different aspects related to the dimension and depth of a binomial edge ideal.  ... 
doi:10.1090/proc/13241 fatcat:jbyiuwq7mjakrahf5tylhrcwq4

On the shortness exponent of 1-tough, maximal planar graphs

Michal Tkáč
1996 Discrete Mathematics  
It is shown that the shortness exponent of the class of l-tough, maximal planar graphs is at most log, 5.  ...  The non-Hamiltonian, l-tough, maximal planar graph with a minimum number of vertices is presented. 0012-365X/96/$15.00 0 1996-Elsevier Science B.V. All rights reserved SSDI 0012-365X(94)00356-4  ...  Acknowledgements The author would like to thank the referees for making a number of suggestions which resulted in a much improved paper and for pointing out an oversight in one of its earlier versions.  ... 
doi:10.1016/0012-365x(94)00356-n fatcat:j6heulytrrbbxmu26qbbldm4wa

Minimally toughness in special graph classes [article]

Gyula Y. Katona, Kitti Varga
2018 arXiv   pre-print
A graph is minimally t-tough, if the toughness of the graph is t and the deletion of any edge from the graph decreases the toughness.  ...  A graph is called t-tough, if the removal of any cutset S leaves at most |S|/t components. The toughness of a graph is the largest t for which the graph is t-tough.  ...  The research of the second author was supported by National Research, Development and Innovation Office NKFIH, K-124171.  ... 
arXiv:1802.00055v1 fatcat:lyo3gefwvfef5eidytlf2f2r74

Hamiltonian prisms on 5-chordal graphs [article]

Mou Gao
2016 arXiv   pre-print
In this paper, we provide a method to find a Hamiltonian cycle in the prism of a 5-chordal graph, which is (1+ϵ)-tough, with some special conditions.  ...  Corollary 2 . 2 Let G be a (1 + ǫ)-tough 5-chordal graph such that, for every 3 mutual remote edges e 1 , e 2 , e 3 of G,  ...  Preliminaries The toughness of graphs is a concept introduced by Chvátal [6] , when he was doing research on Hamiltonicity of graphs.  ... 
arXiv:1603.01371v1 fatcat:2v77fngjd5eyte753qwwczdnca

Local Topological Toughness and Local Factors

Frank Göring, Gyula Y. Katona
2007 Graphs and Combinatorics  
We prove, that besides this the topological toughness has nearly all known nice properties of Katona's edge-toughness and therefore is worth to be considered.  ...  We localize and strengthen Katona's idea of an edge-toughness to a local topological toughness. We disprove a conjecture of Katona concerning the connection between edgetoughness and factors.  ...  Let N C be the set of graphs not being 1-tough, N E be the set of graphs not being 1-edge-tough and N T be the set of graphs not being topological 1-tough.  ... 
doi:10.1007/s00373-007-0736-x fatcat:wiijwm5uazfi3novrj72b5zj5e

Page 61 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews  
(H-AOS; Budapest) Properties of edge-tough graphs. (English summary) Graphs Combin. 15 (1999), no. 3, 315-325.  ...  It is known that a 2f-tough graph is always t-edge-tough. The author proves that this is tight in the following sense: For any 0 < e < 1, there exists a (27 — €)-tough graph that is not t-edge-tough.  ... 

Properties of minimally t-tough graphs [article]

Gyula Y. Katona, Dániel Soltész, Kitti Varga
2017 arXiv   pre-print
A graph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases the toughness.  ...  Kriesell conjectured that for every minimally 1-tough graph the minimum degree δ(G)=2. We show that in every minimally 1-tough graph δ(G)<n+2/3.  ...  All authors are partially supported by the grant of the National Research, Development and Innovation Office -NKFIH, No. 108947.  ... 
arXiv:1604.02746v2 fatcat:d5eolkm3mvaejf3o565o5t4cb4

Toughness and Delaunay triangulations

Michael B. Dillencourt
1990 Discrete & Computational Geometry  
A graph G is 1-tough if for any set P of vertices, c(G -P) < I GI, where c(G -P) is the number of components of the graph obtained by removing P and all attached edges from G, and I GI is the number of  ...  This property arises in the study of Hamiltonian graphs: all Hamiltonian graphs are 1-tough, but not conversely.  ...  Douglas West and Herbert Edelsbrunner pointed out to me the connection between 1-toughness and perfect matchings.  ... 
doi:10.1007/bf02187810 fatcat:twkcz73tezdmxnygspfhdrdc6u

Graph Connectivity and Binomial Edge Ideals [article]

Arindam Banerjee, Luis Núñez-Betancourt
2016 arXiv   pre-print
We relate homological properties of a binomial edge ideal J_G to invariants that measure the connectivity of a simple graph G.  ...  Specifically, we show if R/J_G is a Cohen-Macaulay ring, then graph toughness of G is exactly 1/2. We also give an inequality between the depth of R/J_G and the vertex-connectivity of G.  ...  The second author gratefully acknowledges the support of the National Science Foundation for support through Grant DMS-1502282.  ... 
arXiv:1605.00314v1 fatcat:3xmhuuaztre4hivlt5jbwuvkrm

Page 3571 of Mathematical Reviews Vol. , Issue 91G [page]

1991 Mathematical Reviews  
Some structural properties involving removable edges are developed, including the fact that in a 3-connected graph of order at least 5, at least two edges of every 3-cycle are removable, as is at least  ...  Carsten Thomassen (DK-TUD) 91g:05085 05C45 05C40 Dawes, Robin W. (3-QEN-C); Rodrigues, Marion G. (3-QEN-C) Properties of 1-tough graphs. J. Combin. Math. Combin. Comput. 7 (1990), 153-159.  ... 

Properties of minimally t-tough graphs

Gyula Y. Katona, Dániel Soltész, Kitti Varga
2018 Discrete Mathematics  
A graph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases the toughness.  ...  Kriesell conjectured that for every minimally 1-tough graph the minimum degree δ(G) = 2. We show that in every minimally 1-tough graph δ(G) ≤ n 3 + 1.  ...  All authors are partially supported by the grant of the National Research, Development and Innovation Office -NKFIH, No. 108947.  ... 
doi:10.1016/j.disc.2017.08.033 fatcat:qjafkqrx35hbnn2sk5z7kpc7t4

Toughness and hamiltonicity in k-trees

Hajo Broersma, Liming Xiong, Kiyoshi Yoshimoto
2007 Discrete Mathematics  
By a result of Chen et al. 18-tough chordal graphs are hamiltonian, and by a result of Bauer et al. there exist nontraceable chordal graphs with toughness arbitrarily close to 7 4 .  ...  We consider toughness conditions that guarantee the existence of a hamiltonian cycle in k-trees, a subclass of the class of chordal graphs.  ...  Thus F + {xv, yv} is a 1-tough spanning 2-tree of G containing every edge of C.  ... 
doi:10.1016/j.disc.2005.11.051 fatcat:ngx4nw6sfvb7jk4jvgcf2jyy4u

Page 8943 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
Although connectivity and edge-connectivity are good measures, many of these graphs have stronger properties.  ...  Summary: “A cut edge in a graph G is an edge whose removal increases the number of connected components of G.  ... 

2-walks in 2-tough 2k2-free graphs [article]

Gao Mou
2014 arXiv   pre-print
In this paper, we prove that in every 2-tough 2K_2-free graph, there is a 2-walk.  ...  On 2K 2 -free graphs We present several structural properties of 2K 2 -free graphs which turn out to be very useful in the proof of the main theorem.  ...  Every 1/(k − 2)-tough graph has a k-walk. In particular every 1-tough graph has a 3-walk.  ... 
arXiv:1408.3380v3 fatcat:pcrwqjzmfrcbhgron6iti3mfsy
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