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Long cycles in graphs with large degree sums and neighborhood unions

J. Van den Heuvel
1996 Journal of Graph Theory  
We present and prove several results concerning the length of longest cycles in 2connected or I-tough graphs with large degree sums.  ...  These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings. 0 1996 John Wiley & Sons, Inc.  ...  LONG CYCLES IN GRAPHS 95 (c) This is an immediate consequence of (a) and (b). I + Set U = X + -Y and W = X --Y .  ... 
doi:10.1002/(sici)1097-0118(199601)21:1<87::aid-jgt12>3.3.co;2-k fatcat:6q5hco7pznhgvhrsjqc2hzv76m

Long cycles and neighborhood union in 1-tough graphs with large degree sums

Vu Dinh Hoa
1998 Discussiones Mathematicae Graph Theory  
Acknowledgement The author would like to thank the referees for their comments and many helpful suggestions to improve the proofs in the paper. Long Cycles and Neighborhood Union in ...  ...  Hoa A number of results have been established concerning long cycles in graphs with large degree sums. For details we refer to a survey [4] and [7] .  ...  For k ≤ α we denote by σ k the minimum value of the degree sum of any k pairwise nonadjacent vertices and by N C k (G) the minimum cardinality of the neighborhood union of any k such vertices.  ... 
doi:10.7151/dmgt.1059 fatcat:ngump7kovjgiremjrbl5clr4hm

Long cycles in graphs with large degree sums and neighborhood unions

J. Van den Heuvel
1996 Journal of Graph Theory  
We present and prove several results concerning the length of longest cycles in 2connected or I-tough graphs with large degree sums.  ...  These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings. 0 1996 John Wiley & Sons, Inc.  ...  LONG CYCLES IN GRAPHS 95 (c) This is an immediate consequence of (a) and (b). I + Set U = X + -Y and W = X --Y .  ... 
doi:10.1002/(sici)1097-0118(199601)21:1<87::aid-jgt12>3.0.co;2-p fatcat:bapfkmt2bbaephmymsqcjsw7fe

Page 5926 of Mathematical Reviews Vol. , Issue 92k [page]

1992 Mathematical Reviews  
Various degree and neighborhood conditions are shown to imply long cycles in a graph, and some of these new conditions are shown to generalize well-known conditions for Hamiltonicity of a graph.  ...  a Hamiltonian cycle) and the previous result on neighborhood unions.  ... 

Extremal problems and generalized degrees

Paul Erdős, Ralph J. Faudree, Cecil C. Rousseau
1994 Discrete Mathematics  
(generalized independent degree 6:(G)) is the minimum number of vertices in the union of the neighborhoods of a set of t (independent) vertices of the graph G.  ...  For a positive integer n and a graphical property P, g,( P) will denote the minimal number of edges in a graph G of order n that satisfies property P, and m,(P) will denote the class of graphs with n vertices  ...  Faudree would like to thank the Computer and Automation Institute and the Mathematical Institute of the Hungarian Academy of Sciences for their hospitality during the preparation of this manuscript.  ... 
doi:10.1016/0012-365x(92)00473-5 fatcat:633hkadtyfcd5mrlsqctfvhemi

Page 3206 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
In this paper, the above results are generalised by considering only one big degree vertex or a pair of nonadjacent vertices with larger degree sum.  ...  Summary: “Let G be a 2-connected simple graph with order n (2 > 6) and minimum degree 6.  ... 

A Sub-Graph Expansion-Contraction Method for Error Floor Computation

Nithin Raveendran, David Declercq, Bane Vasic
2020 IEEE Transactions on Communications  
Given the Tanner graph G of a code, and the decoding algorithm D, the method starts from a list of short cycles in G, and expands each cycle by including its sufficiently large neighborhood in G.  ...  Variable nodes of the expanded sub-graphs GEXP are then corrupted exhaustively by all possible error patterns, and decoded by D operating on GEXP.  ...  TSs are identified from sufficiently large neighborhood of short cycles in an actual Tanner graph, i.e. not isolated from their neighborhood.  ... 
doi:10.1109/tcomm.2020.2988676 fatcat:fvv6rggwxngojmkv6gmus2fqrm

Page 1302 of Mathematical Reviews Vol. , Issue 91C [page]

1991 Mathematical Reviews  
R. (4-NOTT) Maximal matchings in graphs with large neighborhoods of independent vertices. J. Graph Theory 14 (1990), no. 2, 167-171.  ...  in G has at least s vertices in the union of their neighborhoods).  ... 

Page 6460 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
(NL-TWENA; Enschede) Long cycles in graphs with large degree sums and neighborhood unions. (English summary) J. Graph Theory 21 (1996), no. 1, 87-102.  ...  In this paper the author gives several lower bounds for the cir- cumference of 2-connected or 1-tough graphs with large degree sums.  ... 

Small Cycles in Small Worlds [article]

Petra M. Gleiss, Peter F. Stadler, Andreas Wagner, David A. Fell
2000 arXiv   pre-print
triangles and a deficit in large cycles.  ...  We characterize the distributions of short cycles in a large metabolic network previously shown to have small world characteristics and a power law degree distribution.  ...  triangles and a deficit in large cycles.  ... 
arXiv:cond-mat/0009124v1 fatcat:jldzoiundvbafoiwywblkyytoe

On the minimal length of the longest trail in a fixed edge-density graph

Vajk Szécsi
2013 Open Mathematics  
AbstractA nearly sharp lower bound on the length of the longest trail in a graph on n vertices and average degree k is given provided the graph is dense enough (k ≥ 12.5).  ...  Eliminate cycles one by one in H by taking a cycle C and dropping its edges out of H (and replace H with the new graph).  ...  Therefore it can be assumed that in A there are no edges in the union of the neighborhood of the three vertices outside A. This gives an empty graph in A with at least 3( /2−2) 2 vertices.  ... 
doi:10.2478/s11533-013-0285-x fatcat:gvpt2ob2ffd6lejbhi5un5swyq

Explicit expanders of every degree and size [article]

Noga Alon
2020 arXiv   pre-print
The spectral analysis relies on the delocalization of eigenvectors of regular graphs in cycle-free neighborhoods.  ...  For any d=p+2 with p ≡ 1 4 prime and all sufficiently large n, we describe a strongly explicit construction of an (n,d, λ)-graph (on exactly n vertices) with λ≤√(2(d-1)) + √(d-2) +o(1) (< (1+√(2)) √(d-  ...  Acknowledgment I thank László Babai, Oded Goldreich, Ryan O'Donnell, Ori Parzanchevski and Peter Sarnak for helpful discussions.  ... 
arXiv:2003.11673v1 fatcat:ep7kjk5ufnd73m4tyxtd2azjhm

LargeP4-free graphs with bounded degree

Myung S. Chung, Douglas B. West
1993 Journal of Graph Theory  
Let ex * (D; H ) denote the maximum number of edges in a connected graph with maximum degree D and no induced subgraph isomorphic to H.  ...  graph G with maximum degree at most D.  ...  ACKNOWLEDGMENT This research was supported in part by ONR Grant N00014-85K0570 and by the University of Illinois Research Board.  ... 
doi:10.1002/jgt.3190170111 fatcat:wpcdqjfxvzf6jbpwmc2r4ofdp4

Decomposing Random Graphs into Few Cycles and Edges

DÁNIEL KORÁNDI, MICHAEL KRIVELEVICH, BENNY SUDAKOV
2014 Combinatorics, probability & computing  
In this paper we show that for most edge probabilitiesG(n, p) can be decomposed into a union ofn/4 +np/2 +o(n) cycles and edges w.h.p. This result is asymptotically tight.  ...  Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph onnvertices can be decomposed intoO(n) cycles and edges.  ...  Conlon and J. Fox for stimulating discussions of the problem treated in this paper.  ... 
doi:10.1017/s0963548314000844 fatcat:55qn27vqtnaizh3ih43rshzszm

A closure concept based on neighborhood unions of independent triples

H.J. Broersma, I. Schiermeyer
1994 Discrete Mathematics  
We show that a more general concept due to Ainouche and Christofides can be restated in terms of degree-sums of independent triples.  ...  We introduce a closure concept which is based on neighborhood unions of independent triples and which also generalizes the closure concept of Bondy and Chvatal.  ...  So we cannot replace n + AU0 in (2) by n + A,,, in order to obtain a more However, by introducing a new condition and considering cardinalities of neighborhood unions instead of degree-sums, we were able  ... 
doi:10.1016/0012-365x(92)00049-w fatcat:mluxadfp4bft3jmiwviyvxk4mq
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