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Various results on the toughness of graphs

1999
*
Networks
*

We discuss how

doi:10.1002/(sici)1097-0037(199905)33:3<233::aid-net9>3.0.co;2-a
fatcat:f42yab3dwffudi7w4gxg3b57iy
*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*. ...##
###
More Progress on Tough Graphs - The Y2K Report

2002
*
Electronic Notes in Discrete Mathematics
*

We continue our quadrennial survey

doi:10.1016/s1571-0653(04)00055-1
fatcat:72x4yoepdjfwbcjt3rhgrksn7m
*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*. ...##
###
Fracture toughness of railway for higher speed rail corridors in Malaysia

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 ...

##
###
Hamiltonian powers in threshold and arborescent comparability graphs

1999
*
Discrete Mathematics
*

For arborescent comparability

doi:10.1016/s0012-365x(98)00346-x
fatcat:llno563brrehdfdyt5s2jk6kei
*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. ...##
###
Graph toughness from Laplacian eigenvalues

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. ...

##
###
Tough graphs and hamiltonian circuits

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*. ...

##
###
Graph connectivity and binomial edge ideals

2016
*
Proceedings of the American Mathematical Society
*

Specifically, we show if R/J G is a Cohen-Macaulay ring, then

doi:10.1090/proc/13241
fatcat:jbyiuwq7mjakrahf5tylhrcwq4
*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. ...##
###
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

1997
*
Discrete Mathematics
*

A new lower bound

doi:10.1016/s0012-365x(96)00238-5
fatcat:otbew4qlp5bezc6cvpkcxcncne
*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. ...##
###
Bipartite Toughness andk-Factors in Bipartite Graphs

2008
*
International Journal of Mathematics and Mathematical Sciences
*

We define a new invarianttB(G)in bipartite

doi:10.1155/2008/597408
fatcat:6iavrsubu5dltfysu4kpgfss3a
*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. ...##
###
Toughness in Graphs – A Survey

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. ...

##
###
The complexity of recognizing tough cubic graphs

1997
*
Discrete Applied Mathematics
*

We conclude with some remarks concerning

doi:10.1016/s0166-218x(97)00030-9
fatcat:l34qg6xr25brbm7vzhsreg7rzy
*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*. ...##
###
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

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*. ...

##
###
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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