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

Bo Ning, Long-tu Yuan
2022 arXiv   pre-print
We point out our result on stability in 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.  ...

### Another Algebraic Proof of Bondy's Theorem on Induced Subsets

Andreas Winter
2000 Journal of combinatorial theory. Series A
K Observe that the bound WnÂd X for m in the 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  ...

### Characterizing Extremal Digraphs for Identifying Codes and Extremal Cases of Bondy's Theorem on Induced Subsets

Florent Foucaud, Reza Naserasr, Aline Parreau
2012 Graphs and Combinatorics
Furthermore, by relating this concept to a well known 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  ).  ...  We are now ready to achieve our goal of classifying the extremal set systems in Bondy's theorem when |A| = |X|. Theorem 17.  ...

### A note on extending Bondy's meta-conjecture

Megan Cream, Ronald J. Gould, Kazuhide Hirohata
2017 The Australasian Journal of Combinatorics
We further extend Bondy's meta-conjecture on pancyclic graphs to a meta-conjecture on chorded pancyclic graphs.  ...  Theorem 2. (Bondy, ) 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  .  ...

### Are there Hard Examples for Frege Systems? [chapter]

Maria Luisa Bonet, Samuel R. Buss, Toniann Pitassi
1995 Feasible Mathematics II
It is shown that 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.  ...

### The strongest monotone degree condition for n-connectedness of a graph

F.T Boesch
1974 Journal of combinatorial theory. Series B (Print)
It is now shown that 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.  ...

### Size Multipartite Ramsey Numbers for Small Paths Versus Stripes

Chula Jayawardene, Lilanthi Samarasekara
2016 Annals of Pure and Applied Mathematics
In proving the 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.  ...

### Paths with two blocks in n-chromatic digraphs

L. Addario-Berry, F. Havet, S. Thomassé
2007 Journal of combinatorial theory. Series B (Print)
This problem may be seen as an extension of 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.  ...

### 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.  ...

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

Noriko H. Arai
2000 Theoretical Computer Science
First, we prove that cut-free LK+permutation has polynomial size proofs for nonunique endnode principle, 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  ...

### Preface

Ewa Drgas-Burchardt, Elżbieta Sidorowicz
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  ...

### 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.  ...

### Sufficient degree conditions fork-edge-connectedness of a graph

D. Bauer, S.L. Hakimi, N. Kahl, E. Schmeichel
2009 Networks
Thus the weakly optimal forcibly k-connected function corresponding to 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.  ...

### On cycle lengths in graphs of moderate degree

H. Bencherif Ait-Djafer
1994 Discrete Mathematics
Subsequently, several authors [3, 5, proved results which support Bondy's metaconjecture.  ...  Theorem 1. 8 ( 8 .  ...  The theorem now follows from Theorem 1.9. 0  ...

### A Relationship Between Thomassen's Conjecture and Bondy's Conjecture

Roman Čada, Shuya Chiba, Kenta Ozeki, Petr Vrána, Kiyoshi Yoshimoto
2015 SIAM Journal on Discrete Mathematics
It is known that Thomassen's conjecture implies 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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