Forbidden subgraphs for graphs with (near) perfect matching to be hamiltonian

2020 ◽  
pp. 1-11
Author(s):  
Xiaojing Yang ◽  
Liming Xiong
Keyword(s):  
Perfect Matching ◽  
Forbidden Subgraphs

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 Forbidden subgraphs of graphs uniquely Hamiltonian-connected from a vertex

1998 ◽  
Vol 187 (1-3) ◽  
pp. 281-290 ◽  
Author(s):  
George R.T. Hendry ◽  
C.J. Knickerbocker ◽  
Patti Frazer Lock ◽  
Michael Sheard
Keyword(s):  
Forbidden Subgraphs ◽  
Hamiltonian Connected

scholarly journals Infinitely many equivalent versions of the graceful tree conjecture

2015 ◽  
Vol 9 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Tao-Ming Wang ◽  
Cheng-Chang Yang ◽  
Lih-Hsing Hsu ◽  
Eddie Cheng
Keyword(s):  
Perfect Matching ◽  
Absolute Difference ◽  
Extra Condition ◽  
Graceful Labeling ◽  
The Absolute

A graceful labeling of a graph with q edges is a labeling of its vertices using the integers in [0, q], such that no two vertices are assigned the same label and each edge is uniquely identified by the absolute difference between the labels of its endpoints. The well known Graceful Tree Conjecture (GTC) states that all trees are graceful, and it remains open. It was proved in 1999 by Broersma and Hoede that there is an equivalent conjecture for GTC stating that all trees containing a perfect matching are strongly graceful (graceful with an extra condition). In this paper we extend the above result by showing that there exist infinitely many equivalent versions of the GTC. Moreover we verify these infinitely many equivalent conjectures of GTC for trees of diameter at most 7. Among others we are also able to identify new graceful trees and in particular generalize the ?-construction of Stanton-Zarnke (and later Koh- Rogers-Tan) for building graceful trees through two smaller given graceful trees.

scholarly journals On the Number of Perfect Matchings in Random Lifts

2010 ◽  
Vol 19 (5-6) ◽  
pp. 791-817 ◽  
Author(s):  
CATHERINE GREENHILL ◽  
SVANTE JANSON ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Asymptotic Formula ◽  
Complete Graph ◽  
Pairwise Disjoint ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Lattice Points ◽  
Laplace’S Method ◽  
Second Moment ◽  
Multiple Dimensions ◽  
Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

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