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Long cycles in graphs with large degree sums and neighborhood unions

1996
*
Journal of Graph Theory
*

We present

doi:10.1002/(sici)1097-0118(199601)21:1<87::aid-jgt12>3.3.co;2-k
fatcat:6q5hco7pznhgvhrsjqc2hzv76m
*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 . ...##
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Long cycles and neighborhood union in 1-tough graphs with large degree sums

1998
*
Discussiones Mathematicae Graph Theory
*

Acknowledgement The author would like to thank the referees for their comments

doi:10.7151/dmgt.1059
fatcat:ngump7kovjgiremjrbl5clr4hm
*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. ...##
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Long cycles in graphs with large degree sums and neighborhood unions

1996
*
Journal of Graph Theory
*

We present

doi:10.1002/(sici)1097-0118(199601)21:1<87::aid-jgt12>3.0.co;2-p
fatcat:bapfkmt2bbaephmymsqcjsw7fe
*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 . ...##
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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*. ...##
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Extremal problems and generalized degrees

1994
*
Discrete Mathematics
*

(generalized independent

doi:10.1016/0012-365x(92)00473-5
fatcat:633hkadtyfcd5mrlsqctfvhemi
*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. ...##
###
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. ...

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

2020
*
IEEE Transactions on Communications
*

Given the Tanner

doi:10.1109/tcomm.2020.2988676
fatcat:fvv6rggwxngojmkv6gmus2fqrm
*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*. ...##
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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*. ...##
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Small Cycles in Small Worlds
[article]

2000
*
arXiv
*
pre-print

triangles

arXiv:cond-mat/0009124v1
fatcat:jldzoiundvbafoiwywblkyytoe
*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*. ...##
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On the minimal length of the longest trail in a fixed edge-density graph

2013
*
Open Mathematics
*

AbstractA nearly sharp lower bound on the length of the longest trail

doi:10.2478/s11533-013-0285-x
fatcat:gvpt2ob2ffd6lejbhi5un5swyq
*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. ...##
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Explicit expanders of every degree and size
[article]

2020
*
arXiv
*
pre-print

The spectral analysis relies on the delocalization of eigenvectors of regular

arXiv:2003.11673v1
fatcat:ep7kjk5ufnd73m4tyxtd2azjhm
*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. ...##
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LargeP4-free graphs with bounded degree

1993
*
Journal of Graph Theory
*

Let ex * (D; H ) denote the maximum number of edges

doi:10.1002/jgt.3190170111
fatcat:wpcdqjfxvzf6jbpwmc2r4ofdp4
*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. ...##
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Decomposing Random Graphs into Few Cycles and Edges

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. ...

##
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A closure concept based on neighborhood unions of independent triples

1994
*
Discrete Mathematics
*

We show that a more general concept due to Ainouche

doi:10.1016/0012-365x(92)00049-w
fatcat:mluxadfp4bft3jmiwviyvxk4mq
*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 ...
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