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Stability in Bondy's theorem on paths and cycles
[article]

2022
*
arXiv
*
pre-print

We point out our result on stability in

arXiv:2207.13650v1
fatcat:r2avjzhn5jfz7jartwucd3pqlm
*Bondy's**theorem*can directly imply a positive solution to the following problem: Is there a polynomial time algorithm to decide whether a 2-connected graph G on ... In this paper, we study the stability result of a well-known*theorem*of Bondy. ... The humble goal of this paper is to prove a stability result of*Bondy's**theorem*, which also improves Voss'*theorem*. Our work is motivated by stability results from extremal graph theory. ...##
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Another Algebraic Proof of Bondy's Theorem on Induced Subsets

2000
*
Journal of combinatorial theory. Series A
*

K Observe that the bound WnÂd X for m in the

doi:10.1006/jcta.1999.3013
fatcat:oe5ms72o75aspcqw6najcrrvdy
*theorem*is optimal: for 2nÂ3<d<n this is trivial, as a code of minimal distance d can have no more than two elements. ... For d>1 we can iterate*Theorem*2 and obtain A 0 =< A j =[1+d( j&1), ACKNOWLEDGMENT Thanks to Levon Khachatrian, who posed Bondy's theorem as an exercise in his lecture on extremal combinatorics, and ...##
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Characterizing Extremal Digraphs for Identifying Codes and Extremal Cases of Bondy's Theorem on Induced Subsets

2012
*
Graphs and Combinatorics
*

Furthermore, by relating this concept to a well known

doi:10.1007/s00373-012-1136-4
fatcat:3tjnap5bobgdph2jswxmg7naey
*theorem*of A. Bondy on set systems we classify the extremal cases for this*theorem*. ... In Section 4, we discuss the relationship between this result and the following*theorem*of A. Bondy on "induced subsets":*Theorem*2 (*Bondy's**theorem*[4] ). ... We are now ready to achieve our goal of classifying the extremal set systems in*Bondy's**theorem*when |A| = |X|.*Theorem*17. ...##
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A note on extending Bondy's meta-conjecture

2017
*
The Australasian Journal of Combinatorics
*

We further extend

dblp:journals/ajc/CreamGH17
fatcat:uh64zvgfozdf5bby7phbpybr3i
*Bondy's*meta-conjecture on pancyclic graphs to a meta-conjecture on chorded pancyclic graphs. ...*Theorem*2. (Bondy, [1]) If G is a graph of order n ≥ 3 with σ 2 (G) ≥ n, then G is pancyclic or G = K n/2,n/2 . In this paper, we wish to extend*Bondy's*meta-conjecture. ... This completes the proof of*Theorem*3. Conclusion We end by making a specific conjecture concerning chorded pancyclic graphs. If true, this would extend another result of*Bondy's*[1] . ...##
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Are there Hard Examples for Frege Systems?
[chapter]

1995
*
Feasible Mathematics II
*

It is shown that

doi:10.1007/978-1-4612-2566-9_3
fatcat:4e2z5jonanfcnj4rqtcwoddjky
*Bondy's**theorem*and a version of the Kruskal-Katona*theorem*actually have polynomial-size Frege proofs. ...*Bondy's**theorem*is shown to have constant-depth, polynomial-size proofs in Frege+PHP, and to be equivalent in I∆0 to the pigeonhole principle. ... In section 4.2, we further discuss Frankl's*theorem*, which is a generalization of*Bondy's**theorem*. ...##
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The strongest monotone degree condition for n-connectedness of a graph

1974
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Journal of combinatorial theory. Series B (Print)
*

It is now shown that

doi:10.1016/0095-8956(74)90058-6
fatcat:ulgqayqcnngpxbzphjrp32idka
*Bondy's*condition is "best" in the above sense.*THEOREM*3. ... It is shown here that*Bondy's**theorem*gives the strongest monotone degree condition for forcible n-connectedness. ...##
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Size Multipartite Ramsey Numbers for Small Paths Versus Stripes

2016
*
Annals of Pure and Applied Mathematics
*

In proving the

doi:10.22457/apam.v12n2a12
fatcat:mkpr3zsrivagfc7cx5o4qydax4
*theorem*given below we will use the following lemma which is a direct consequence of*Bondy's*Lemma. Lemma 3. ... Therefore, by*Bondy's*lemma ] [N G B will contain a blue Hamilton cycle. ...##
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Paths with two blocks in n-chromatic digraphs

2007
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Journal of combinatorial theory. Series B (Print)
*

This problem may be seen as an extension of

doi:10.1016/j.jctb.2006.10.001
fatcat:vp2sn7c5mrafxgzz2l3fgkkd5e
*Bondy's**theorem*(*Theorem*1) which proves this statement for directed cycles (l = 0). ... Thus by*Bondy's**theorem*, D 1 contains a circuit of length at least k but shorter than C. ...##
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Page 28 of Mathematical Reviews Vol. , Issue 96a
[page]

1996
*
Mathematical Reviews
*

*Bondy’s*

*theorem*[J. Combin. Theory Ser. ...

*Bondy’s*

*theorem*is shown to have constant-depth, polynomial-size proofs in Frege + pigeonhole principle, and to be equivalent in [Ap to the pigeonhole principle. ...

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Tractability of Cut-free Gentzen-type propositional calculus with permutation inference II

2000
*
Theoretical Computer Science
*

First, we prove that cut-free LK+permutation has polynomial size proofs for nonunique endnode principle,

doi:10.1016/s0304-3975(98)00186-8
fatcat:hsf5mmthhva4zdqb52chl3srgq
*Bondy's**theorem*. ... Second, we remark the fact that permutation inference has an advantage over renaming inference in automated*theorem*proving, since GCNF+renaming does not always satisfy the feasible subformula property ...*Bondy's**theorem**Bondy's**theorem*states that in any n×n (0,1)-matrix containing n pairwise distinct rows, there exists a column such that, if the column is deleted, the resulting (n − 1)×n matrix still ...##
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Preface

2015
*
Discussiones Mathematicae Graph Theory
*

*Theorem*on Sufficient Conditions for Hamiltonian Properties Jaroslav Nešetřil (Charles University, Czech Republic) Sparsity: Graph Algorithms and Structure Gabriel Semanišin (P.J.Šafárik University, Slovakia ... Well-Covered Planar Triangulations Michael Henning (University of Johannesburg, Republic of South Africa) Total Domination in Graphs Arnfried Kemnitz (Technische Universitaet Braunschweig, Germany) Generalizing

*Bondy's*...

##
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Page 6636 of Mathematical Reviews Vol. , Issue 92m
[page]

1992
*
Mathematical Reviews
*

*Bondy’s*condition (d(x) +d(y)+d(z) > (3n—2)/2 for inde- pendent 3-sets x, y, z) for Hamiltonian cycles in undirected graphs, but the simple transformation above does not immediately yield

*Bondy’s*

*theorem*... The author leaves us with two conjectures that would improve the

*theorem*. ...

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Sufficient degree conditions fork-edge-connectedness of a graph

2009
*
Networks
*

Thus the weakly optimal forcibly k-connected function corresponding to

doi:10.1002/net.20299
fatcat:ocgrbdveefal7akktd7obvrufm
*Bondy's*condition (*Theorem*1.2) is the strongest monotone increasing forcibly k-connected function. ... This proves*Theorem*3.1. ... We now show that*Theorem*4.1 is at least as strong as*Theorem*1.2 (*Bondy's*condition), and sometimes significantly better. We first prove the following. ...##
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On cycle lengths in graphs of moderate degree

1994
*
Discrete Mathematics
*

Subsequently, several authors [3, 5, proved results which support

doi:10.1016/0012-365x(94)90143-0
fatcat:bid2afpcy5aini73w4teaevsbu
*Bondy's*metaconjecture. ...*Theorem*1. 8 ( 8 . ... The*theorem*now follows from*Theorem*1.9. 0 ...##
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A Relationship Between Thomassen's Conjecture and Bondy's Conjecture

2015
*
SIAM Journal on Discrete Mathematics
*

It is known that Thomassen's conjecture implies

doi:10.1137/130937974
fatcat:qs4y4e7axjhzjgclbwd6ka733m
*Bondy's*conjecture, but nothing about the converse has been shown. ... In this paper, we show that*Bondy's*conjecture implies a slightly weaker version of Thomassen's conjecture: every 4-connected line graph with minimum degree at least 5 has a Hamiltonian cycle. ... Hence*Theorems*10 and 11 imply*Theorem*5. We will show*Theorems*10 and 11 in Sections 4 and 5, respectively. Proof of*Theorem*10 Suppose that Conjecture 2 is true. ...
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