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Minimum Degree Conditions for Cycles Including Specified Sets of Vertices

2004
*
Graphs and Combinatorics
*

This paper generalises the concept

doi:10.1007/s00373-004-0565-0
fatcat:uioxc5aajffxniyorefiy2rsda
*of*vertex pancyclic graphs. We define a graph as*set*-pancyclic if*for*every*set*S*of**vertices*there is a*cycle**of*every possible length containing S. ... We show that if the*minimum**degree**of*a graph exceeds half its order then the graph is*set*-pancyclic. ... Much has been written about generalisations*of*this*condition*, by*for*example, replacing the*minimum**degree**condition*with*degree**conditions*on*sets**of**vertices*. ...##
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Page 6414 of Mathematical Reviews Vol. , Issue 91M
[page]

1991
*
Mathematical Reviews
*

This paper considers a generalization in which the

*specified**set**includes*both*vertices*and edges; namely, given a connected, unidirected graph G (without pendant*vertices*) and a*specified**set*A*of**vertices*... The problem*of*finding a*cycle*in a given graph which passes through a*specified**set**of**vertices*has been around*for*a long time. ...##
###
A look at cycles containing specified elements of a graph

2009
*
Discrete Mathematics
*

Here the natural choices are a

doi:10.1016/j.disc.2008.04.017
fatcat:uzmzlcs6lvdzjaf5nd55wmysjm
*set**of**vertices**of**specified*size, a*set**of**vertices*satisfying a property (like all*vertices**of*maximum*degree*or all*vertices**of**degree*at least k), a*set**of*independent ... Increasing connectivity beyond some absolute*minimum*value (often 2-connectedness) may allow us to relax other*conditions*. 3. The type*of**specified*elements to be*included*in the*cycle*. ... Acknowledgements The author would like to thank the referees*for*their very helpful comments. ...##
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Page 5297 of Mathematical Reviews Vol. , Issue 93j
[page]

1993
*
Mathematical Reviews
*

After surveying sufficient

*conditions**for*connectivity or edge-connectivity to equal*minimum**degree*, the author proves a sufficient*condition**for*super- A in terms*of*the diameter D, order n,*minimum**degree*... Necessary and sufficient*conditions*on the*set**of**degrees*are given, and algorithms are developed*for*the construction*of*the realisable graphs. ...##
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On low degree k-ordered graphs

2008
*
Discrete Mathematics
*

these k

doi:10.1016/j.disc.2007.05.024
fatcat:dbdj7njg2nadleksnxaml7p7v4
*vertices*in the*specified*order. ... A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if,*for*any sequence*of*k distinct*vertices*v 1 , . . . , v k*of*G, there exists a*cycle*(respectively, a hamiltonian*cycle*) in G containing ... Acknowledgments This research was performed at the University*of*Minnesota Duluth under the supervision*of*Professor Joseph A. Gallian. ...##
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On degree sum conditions for long cycles and cycles through specified vertices

2008
*
Discrete Mathematics
*

This suggests that the

doi:10.1016/j.disc.2007.10.048
fatcat:j6352k3p5zah5jzocsbn77i45q
*degree*sum*of*nonadjacent two*vertices*is important*for*guaranteeing the existence*of*these*cycles*. ... If ∆ 2 (S) ≥ |V (G)|*for*every independent*set*S*of*order κ(X ) + 1 in G[X ], then G has a*cycle*that*includes*every vertex*of*X . ... Acknowledgments The author would like to thank Professor Akira Saito*for*stimulating discussions and important suggestions. ...##
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Scheduling Automated Traffic on a Network of Roads

2006
*
IEEE Transactions on Vehicular Technology
*

Each vehicle has a

doi:10.1109/tvt.2006.877472
fatcat:jsbbknjswbgldfxpma533dxde4
*specified*route from its origin to its destination, and the task*of*the scheduler is to provide timed trajectories*for*all vehicles, which follow the respective vehicles' routes and ... Our main result is a sufficient*condition*on the graph*of*the road network and on the initial distribution*of*vehicles, under which there exists a scheduling algorithm that is guaranteed to clear the system ... The*minimum*time to clear the system is therefore the*minimum*number*of*such subsets that together*include*all the vehicles, which corresponds to the*minimum*number*of*disjoint independent*sets*that cover ...##
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Page 4716 of Mathematical Reviews Vol. , Issue 93i
[page]

1993
*
Mathematical Reviews
*

Math. 14 (1966), 778-781; MR 34 #7400], the authors provide a survey

*of**conditions*that guarantee that the*minimum**degree*6*of*a graph is equal to its edge connectivity 4. This*includes*J. ... In this paper sufficient*conditions**for*a digraph to be super-A are given in terms*of*parameters such as diameter and*minimum**degree*. ...##
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Page 8943 of Mathematical Reviews Vol. , Issue 2003m
[page]

2003
*
Mathematical Reviews
*

to have 2-factors with

*cycles*through*specified**vertices*. ... Summary: “In this paper we present some new sufficient*conditions**for*equality*of*edge-connectivity and*minimum**degree**of*graphs and digraphs as well as*of*bipartite graphs and digraphs.” 2003m:05110 05C40 ...##
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Placing Specified Vertices at Precise Locations on a Hamiltonian Cycle

2017
*
Graphs and Combinatorics
*

Sharp

doi:10.1007/s00373-017-1762-y
fatcat:2hd2xh7tobhjbpgwflswz5hv4m
*minimum**degree*and*degree*sum*conditions*are proven*for*the existence*of*a Hamiltonian*cycle*passing through*specified**vertices*with prescribed distances between them in large graphs. ... Our next result provides a*degree*sum*condition**for*the same placement*of**specified**vertices*on a Hamiltonian*cycle*. ... Faudree and Gould recently provided a sharp*minimum**degree**condition**for*the placement*of**specified**vertices*at precise locations relative to each other on a Hamiltonian*cycle*. ...##
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On low degree k-ordered graphs
[article]

2005
*
arXiv
*
pre-print

these k

arXiv:math/0509411v1
fatcat:quahv66h2rh57p76rd2zjziyym
*vertices*in the*specified*order. ... A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if,*for*any sequence*of*k distinct*vertices*v_1, ..., v_k*of*G, there exists a*cycle*(respectively, a hamiltonian*cycle*) in G containing ... The author would like to thank Professor Gallian*for*his support and encouragement as well as Denis Chebikin, Philip Matchett and Melanie Wood*for*many useful suggestions. ...##
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Degree sum and vertex dominating paths

2018
*
Journal of Graph Theory
*

T is a

doi:10.1002/jgt.22249
fatcat:k25kr2jlb5ecdpg5dzeunbfq2m
*minimum*dominating*set**of**vertices*. ... The main result is a proof that every n-vertex k-connected graph such that σ 2 (G) ≥ 2n k+2 + f (k) contains a path*of*length at most O(|T |), through any*set**of*T*vertices*where |T | = o(n). ... The next lemma provides*conditions*under which the existence*of*some*cycle*through a*specified**set**of**vertices*implies the existence*of*a small*cycle*through the*specified**set*. Lemma 3.4. ...##
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A Simple Extension of Dirac's Theorem on Hamiltonicity
[article]

2016
*
arXiv
*
pre-print

The classical Dirac theorem asserts that every graph G on n

arXiv:1606.03687v1
fatcat:zdavh5hx7fa6za6pd2f7vodeum
*vertices*with*minimum**degree*δ(G) > n/2 is Hamiltonian. The lower bound*of*n/2 on the*minimum**degree**of*a graph is tight. ... Finally, we present a self-contained proof*for*our algorithm which provides insight into the structure*of*Hamiltonian*cycles*when δ(G) > n/2 and is promising*for*extending the results*of*this paper to ... An important sufficient*condition**for*Hamiltonicity proved in 1952 by Dirac [2] is that every graph on n*vertices*with*minimum**degree*at least n/2 is Hamiltonian. ...##
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Locating pairs of vertices on Hamiltonian cycles

2012
*
Discrete Mathematics
*

Given a fixed positive integer k ≥ 2 and a fixed pair

doi:10.1016/j.disc.2012.02.006
fatcat:4ieoojaycjgszenc6cnv77demm
*of**vertices*x and y in a graph*of*sufficiently large order n = n(k),*minimum**degree**conditions*that imply the existence*of*a Hamiltonian*cycle*C such ... that the distance on the*cycle*C between x and y is precisely k will be proved. ... Acknowledgments The authors would like to thank the referees*for*their careful reading*of*the manuscript and their valuable suggestions. ...##
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A new algorithm to find fuzzy Hamilton cycle in a fuzzy network using adjacency matrix and minimum vertex degree

2016
*
SpringerPlus
*

In this paper, a new algorithm is proposed to find fuzzy Hamiltonian

doi:10.1186/s40064-016-3473-x
pmid:27818892
pmcid:PMC5075334
fatcat:f6ds2hegizfojncq37sftc7yia
*cycle*using adjacency matrix and the*degree**of*the*vertices**of*a fuzzy graph. ... This leads to the development*of*new algorithms and new theorems. A graph is a pictorial representation*for*a*set**of*objects in which some pairs are connected by links. ... Acknowledgements The authors would like to thank the editor and anonymous referees*of*this journal*for*their constructive suggestions. ...
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