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Various results on the toughness of graphs

Hajo Broersma, Erik Engbers, Huib Trommel
1999 Networks  
We discuss how the toughness of (spanning) subgraphs of G and related graphs depends on τ (G), we give some sufficient (degree) conditions implying τ (G) ≥ t, and we study which subdivisions of 2-connected  ...  graphs have minimally 2-tough squares.  ...  The results in the other sections are based on an attempt to get more grip on the concept of toughness.  ... 
doi:10.1002/(sici)1097-0037(199905)33:3<233::aid-net9>3.0.co;2-a fatcat:f42yab3dwffudi7w4gxg3b57iy

More Progress on Tough Graphs - The Y2K Report

Doug Bauer, Hajo Broersma, Edward Schmeichel
2002 Electronic Notes in Discrete Mathematics  
We continue our quadrennial survey of results that relate the toughness of a graph to its cycle structure.  ...  We now know that not every 2-tough graph is hamiltonian. In fact for every > 0, there exists a (9/4− ) -tough nontraceable graph.  ...  It is natural to inquire whether the problem of recognizing t-tough graphs remains NP-hard for various subclasses of graphs.  ... 
doi:10.1016/s1571-0653(04)00055-1 fatcat:72x4yoepdjfwbcjt3rhgrksn7m

Fracture toughness of railway for higher speed rail corridors in Malaysia

M A Rojan, M S M Azmi, M S Abdul Majid, R Daud, K S Basaruddin
2019 IOP Conference Series: Materials Science and Engineering  
The fracture toughness of the steel was found to be 44.009 MPa√m, which is within the range of the various rail steels used around the world..  ...  The aim of this study was to evaluate the fracture toughness of rail steel.  ...  Acknowledgement The author would like to acknowledge the support from the Fundamental Research Grant Scheme (FRGS) under a grant number of FRGS/1/2018/TK03/UNIMAP/03/6 from the Ministry of Education Malaysia  ... 
doi:10.1088/1757-899x/670/1/012065 fatcat:xbhf2b5nnvcpboe2dbdfykg4hy

Hamiltonian powers in threshold and arborescent comparability graphs

Sam Donnelly, Garth Isaak
1999 Discrete Mathematics  
For arborescent comparability graphs we have similar results but also show that for one type of completion problem an 'obvious' minimax condition fails.  ...  For threshold graphs we give additional necessary and sufficient conditions in terms of vertex degrees as well as a minimax formula for the length of a longest cycle power.  ...  Acknowledgements Thanks to a referee for a very careful reading of the manuscript and useful comments.  ... 
doi:10.1016/s0012-365x(98)00346-x fatcat:llno563brrehdfdyt5s2jk6kei

Graph toughness from Laplacian eigenvalues

Xiaofeng Gu, Willem H. Haemers
2022 Algebraic Combinatorics  
The toughness t(G) of a graph G = (V, E) is defined as t(G) = min |S| c(G−S) , in which the minimum is taken over all S ⊂ V such that G − S is disconnected, where c(G − S) denotes the number of components  ...  The toughness t(G) of a graph G is defined as t(G) = min{ |S| c(G−S) }, where the minimum is taken over all proper subsets S ⊂ V such that c(G − S) > 1.  ...  The first author is partially supported by a grant from the Simons Foundation (522728, XG). Acknowledgements.  ... 
doi:10.5802/alco.197 fatcat:avkoi45mlveehc5syp2v352nry

Tough graphs and hamiltonian circuits

V. Chvátal
1973 Discrete Mathematics  
The toughness of a graph G is defined as the largest real number: t such that deletion of any s points from 6 results in a graph which is either connected or elst: has at most s/t components.  ...  Since a square of a k-connectc;d graph is always ktough., a proof of this conjecture with to = 2 would imply Fleischner's thec,rem (the square of a block is hamiltonian).  ...  Introduction In this paper, we introduce a new invariant for graph:;. Ijt measures in a simple way how tightly various pieces of a graph 'hold together; therefore we call it toughness.  ... 
doi:10.1016/0012-365x(73)90138-6 fatcat:vexd4udggbgrpfs4q5vfvf5uvm

Graph connectivity and binomial edge ideals

Arindam Banerjee, Luis Núñez-Betancourt
2016 Proceedings of the American Mathematical Society  
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.  ...  We relate homological properties of a binomial edge ideal J G to invariants that measure the connectivity of a simple graph G.  ...  Acknowledgments The authors thank Craig Huneke for helpful discussions. The authors are also grateful to the anonymous referee for valuable comments and suggestions.  ... 
doi:10.1090/proc/13241 fatcat:jbyiuwq7mjakrahf5tylhrcwq4

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

1991 Mathematical Reviews  
The focus of this paper is on nonremovable edges, and the main result states that every 3-connected graph of order n > 5 has at most (4n — 5)/3 nonremovable edges.  ...  For v > 4 the author determines the largest number f(v) such that every simple 3-connected graph on v vertices has f(v) edge contractions which result in a smaller 3-connected graph.  ... 

Maximum and minimum toughness of graphs of small genus

Wayne Goddard, Michael D. Plummer, Henda C. Swart
1997 Discrete Mathematics  
A new lower bound on the toughness t(G) of a graph G in terms of its connectivity ~c(G) and genus 7(G) is obtained.  ...  For 7 > 0, the bound is sharp via an infinite class of extremal graphs all of girth 4. For planar graphs, the bound is t(G) > ~c(G)/2 -1.  ...  Relations linking toughness to various forms of hamiltonicity and to the existence of k-factors in graphs have been widely investigated and will not be discussed here.  ... 
doi:10.1016/s0012-365x(96)00238-5 fatcat:otbew4qlp5bezc6cvpkcxcncne

Bipartite Toughness andk-Factors in Bipartite Graphs

Guizhen Liu, Jianbo Qian, Jonathan Z. Sun, Rui Xu
2008 International Journal of Mathematics and Mathematical Sciences  
We define a new invarianttB(G)in bipartite graphs that is analogous to the toughnesst(G)and we give sufficient conditions in term oftB(G)for the existence ofk-factors in bipartite graphs.  ...  We also show that these results are sharp.  ...  Acknowledgment The first author's work is partially supported by National Natural Science Foundation of China NSFC 10871119.  ... 
doi:10.1155/2008/597408 fatcat:6iavrsubu5dltfysu4kpgfss3a

Toughness in Graphs – A Survey

Douglas Bauer, Hajo Broersma, Edward Schmeichel
2006 Graphs and Combinatorics  
Toughness and Circumference In this section we survey results concerning the relationship between the toughness of a graph and its circumference.  ...  These categories are circumference, the disproof of the 2-tough conjecture, factors, special graph classes, computational complexity, and miscellaneous results as they relate to toughness.  ...  Acknowledgements We gratefully acknowledge the referees for carefully reading this survey.  ... 
doi:10.1007/s00373-006-0649-0 fatcat:pybxyua33rehpmiybdbj2cnzru

The complexity of recognizing tough cubic graphs

D. Bauer, J. van den Heuvel, A. Morgana, E. Schmeichel
1997 Discrete Applied Mathematics  
We conclude with some remarks concerning the complexity of recognizing certain subclasses of tough graphs.  ...  We show that it is NP-hard to determine if a cubic graph G is l-tough. We then use this result to show that for any integer t > 1, it is NP-hard to determine if a 3 t-regular graph is t-tough.  ...  For any rational t 2 1, t-TOUGH is NP-hard. It seems natural to inquire whether the problem of recognizing t-tough graphs remains NP-hard for various subclasses of graphs.  ... 
doi:10.1016/s0166-218x(97)00030-9 fatcat:l34qg6xr25brbm7vzhsreg7rzy

Page 7563 of Mathematical Reviews Vol. , Issue 98M [page]

1998 Mathematical Reviews  
bound on AO(G) is tight for various graphs including K,, K,,, and trees.  ...  This result improves a result of Faudree et al. that the graph contains all cycles of length m where 3 < m < (n+ 19) under the same condition.  ... 

Tough graphs and hamiltonian circuits

V. Chvátal
2006 Discrete Mathematics  
The toughness of a graph G is defined as the largest real number t such that deletion of any s points from G results in a graph which is either connected or else has at most s/t components.  ...  Obviously, a t-tough graph is s-tough for all s < t. If G is not complete, then there is a largest t such that G is t-tough; this t will be called the toughness of G and denoted by t (G). On the other  ...  Introduction In this paper, we introduce a new invariant for graphs. It measures in a simple way how tightly various pieces of a graph hold together; therefore we call it toughness.  ... 
doi:10.1016/j.disc.2006.03.011 fatcat:ohvy35ngifhqtp4aljnkgrxt74

Page 435 of Mathematical Reviews Vol. 49, Issue 2 [page]

1975 Mathematical Reviews  
Maal for the conjecture of the result (ii). Included in the paper are various bounds for 6(G,,), the number of lines in a maximal line bipartite subgraph of a given p point graph G,. L. V.  ...  The path number 7(@) of a graph G is the smallest number of line-disjoint paths that cover ail of G. The authors give simplified proofs of some results on path numbers previously obtained by R. G.  ... 
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