Every 3-connected {K1,3,Z7}-free graph of order at least 21 is Hamilton-connected

2021 ◽  
Vol 344 (6) ◽  
pp. 112350
Author(s):  
Zdeněk Ryjáček ◽  
Petr Vrána
Keyword(s):  
Free Graph ◽  
Hamilton Connected

scholarly journals Every 3-connected distance claw-free graph is Hamilton-connected

2003 ◽  
Vol 268 (1-3) ◽  
pp. 185-197 ◽  
Author(s):  
Rao Li ◽  
R.H. Schelp
Keyword(s):  
Free Graph ◽  
Hamilton Connected

scholarly journals Extending Cycles Locally to Hamilton Cycles

10.37236/3960 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Matthias Hamann ◽  
Florian Lehner ◽  
Julian Pott
Keyword(s):  
Unit Circle ◽  
Unit Interval ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Hamilton Cycles ◽  
Free Graph ◽  
Locally Finite ◽  
Hamilton Connected

A Hamilton circle in an infinite graph is a homeomorphic copy of the  unit circle $S^1$ that contains all vertices and all ends precisely once. We prove that every connected, locally connected, locally finite, claw-free graph has such a Hamilton circle, extending a result of Oberly and Sumner to infinite graphs. Furthermore, we show that such graphs are Hamilton-connected if and only if they are $3$-connected, extending a result of Asratian. Hamilton-connected means that between any two vertices there is a Hamilton arc, a homeomorphic copy of the unit interval $[0,1]$ that contains all vertices and all ends precisely once.

scholarly journals Toughness, Forbidden Subgraphs and Pancyclicity

2021 ◽  
Vol 37 (3) ◽  
pp. 839-866
Author(s):  
Wei Zheng ◽  
Hajo Broersma ◽  
Ligong Wang
Keyword(s):  
Recent Article ◽  
Forbidden Subgraph ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Answer ◽  
Open Case

AbstractMotivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

scholarly journals Universal and unavoidable graphs

2021 ◽  
pp. 1-14
Author(s):  
Matija Bucić ◽  
Nemanja Draganić ◽  
Benny Sudakov
Keyword(s):  
Free Graph ◽  
Turán Number ◽  
Work Done

Abstract The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.

scholarly journals The average tree solution for cycle-free graph games

2008 ◽  
Vol 62 (1) ◽  
pp. 77-92 ◽  
Author(s):  
P. Jean Jacques Herings ◽  
Gerard van der Laan ◽  
Dolf Talman
Keyword(s):  
Free Graph ◽  
Graph Games

Every 3-connected {K 1,3,N 3,3,3}-free graph is Hamiltonian

2013 ◽  
Vol 56 (8) ◽  
pp. 1585-1595 ◽  
Author(s):  
HouYuan Lin ◽  
ZhiQuan Hu
Keyword(s):  
Free Graph

Mixed Ramsey Numbers Revisited

2003 ◽  
Vol 12 (5-6) ◽  
pp. 653-660
Author(s):  
C. C. Rousseau ◽  
S. E. Speed
Keyword(s):  
Asymptotic Behaviour ◽  
Ramsey Number ◽  
Open Problems ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Dense Graphs

Given a graph Hwith no isolates, the (generalized) mixed Ramsey number is the smallest integer r such that every H-free graph of order r contains an m-element irredundant set. We consider some questions concerning the asymptotic behaviour of this number (i) with H fixed and , (ii) with m fixed and a sequence of dense graphs, in particular for the sequence . Open problems are mentioned throughout the paper.

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