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Minimum Degree Conditions for Cycles Including Specified Sets of Vertices

Wayne Goddard
2004 Graphs and Combinatorics
This paper generalises the concept 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.  ...

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

Ronald J. Gould
2009 Discrete Mathematics
Here the natural choices are a 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.  ...

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.  ...

On low degree k-ordered graphs

Karola Mészáros
2008 Discrete Mathematics
these k 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.  ...

On degree sum conditions for long cycles and cycles through specified vertices

Tomoki Yamashita
2008 Discrete Mathematics
This suggests that the 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.  ...

Scheduling Automated Traffic on a Network of Roads

A. Giridhar, P.R. Kumar
2006 IEEE Transactions on Vehicular Technology
Each vehicle has a 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  ...

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.  ...

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  ...

Placing Specified Vertices at Precise Locations on a Hamiltonian Cycle

Ronald J. Gould, Colton Magnant, Pouria Salehi Nowbandegani
2017 Graphs and Combinatorics
Sharp 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.  ...

On low degree k-ordered graphs [article]

Karola Meszaros
2005 arXiv   pre-print
these k 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.  ...

Degree sum and vertex dominating paths

Jill Faudree, Ralph J. Faudree, Ronald J. Gould, Paul Horn, Michael S. Jacobson
2018 Journal of Graph Theory
T is a 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.  ...

A Simple Extension of Dirac's Theorem on Hamiltonicity [article]

Yasemin Büyükçolak and Didem Gözüpek and Sibel Özkan and Mordechai Shalom
2016 arXiv   pre-print
The classical Dirac theorem asserts that every graph G on n 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.  ...

Locating pairs of vertices on Hamiltonian cycles

Ralph J. Faudree, Hao Li
2012 Discrete Mathematics
Given a fixed positive integer k ≥ 2 and a fixed pair 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.  ...

A new algorithm to find fuzzy Hamilton cycle in a fuzzy network using adjacency matrix and minimum vertex degree

A. Nagoor Gani, S. R. Latha
2016 SpringerPlus
In this paper, a new algorithm is proposed to find fuzzy Hamiltonian 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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