scholarly journals Vertex-Oriented Hamilton Cycles in Directed Graphs

10.37236/204 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Michael J. Plantholt ◽  
Shailesh K. Tipnis
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Sufficient Conditions ◽  
Hamilton Cycle ◽  
Directed Path ◽  
Hamilton Cycles ◽  
Even Order

Let $D$ be a directed graph of order $n$. An anti-directed Hamilton cycle $H$ in $D$ is a Hamilton cycle in the graph underlying $D$ such that no pair of consecutive arcs in $H$ form a directed path in $D$. We prove that if $D$ is a directed graph with even order $n$ and if the indegree and the outdegree of each vertex of $D$ is at least ${2\over 3}n$ then $D$ contains an anti-directed Hamilton cycle. This improves a bound of Grant. Let $V(D) = P \cup Q$ be a partition of $V(D)$. A $(P,Q)$ vertex-oriented Hamilton cycle in $D$ is a Hamilton cycle $H$ in the graph underlying $D$ such that for each $v \in P$, consecutive arcs of $H$ incident on $v$ do not form a directed path in $D$, and, for each $v \in Q$, consecutive arcs of $H$ incident on $v$ form a directed path in $D$. We give sufficient conditions for the existence of a $(P,Q)$ vertex-oriented Hamilton cycle in $D$ for the cases when $|P| \geq {2\over 3}n$ and when ${1\over 3}n \leq |P| \leq {2\over 3}n$. This sharpens a bound given by Badheka et al.

A Minimax Equality Related to the Longest Directed Path in an Acyclic Graph

1975 ◽  
Vol 27 (2) ◽  
pp. 348-351 ◽  
Author(s):  
K. Vidyasankar ◽  
D. H. Younger
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Directed Path ◽  
Acyclic Graph ◽  
Vertex Set ◽  
Acyclic Directed Graph

As an analog of a recently established minimax equality for directed graphs [1], I. Simon has suggested that the following be investigated.1.1. For a finite acyclic directed graph G, a minimum collection of directed coboundaries whose union is the edge set of G has cardinality equal to that of a maximum strong matching of G.This minimax equality is here proved, using a characterization of a maximum strong matching of an acyclic graph as the set of edges of a longest directed path in the graph.The terms employed in the above theorem are defined as follows. Let G be a finite directed graph with vertex set VG and edge set eG

On Subtrees of Directed Graphs with No Path of Length Exceeding One

1970 ◽  
Vol 13 (3) ◽  
pp. 329-332 ◽  
Author(s):  
R. L. Graham
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Directed Path ◽  
Directed Tree ◽  
Prove Theorem

The following theorem was conjectured to hold by P. Erdös [1]:Theorem 1. For each finite directed tree T with no directed path of length 2, there exists a constant c(T) such that if G is any directed graph with n vertices and at least c(T)n edges and n is sufficiently large, then T is a subgraph of G. In this note we give a proof of this conjecture. In order to prove Theorem 1, we first need to establish the following weaker result.

scholarly journals Closing Gaps in Problems related to Hamilton Cycles in Random Graphs and Hypergraphs

10.37236/5025 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Asaf Ferber
Keyword(s):  
Random Graph ◽  
Directed Graph ◽  
Random Graphs ◽  
High Probability ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Edge Probability

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for $k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform hypergraph on $n$ vertices, with edge probability $p$, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This generalizes results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K>0$ such for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random graph on $n$ vertices with edge probability $p$, and suppose that its edges are being colored with $n$ colors uniformly at random. Then, w.h.p. the resulting graph contains a Hamilton cycle with for which all the colors appear (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for graphs on an even number of vertices, where for odd $n$ their $p$ was $\omega((\log n)/n)$. Lastly, we show that for $p=(1+o(1))(\log n)/n$, if we randomly color the edge set of a random directed graph $D_{n,p}$ with $(1+o(1))n$ colors, then w.h.p. one can find a rainbow Hamilton cycle where all the edges are directed in the same way.

scholarly journals Hamilton Cycles in Sparse Robustly Expanding Digraphs

10.37236/7418 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Allan Lo ◽  
Viresh Patel
Keyword(s):  
Minimum Degree ◽  
Directed Graphs ◽  
Hamilton Cycle ◽  
Regularity Lemma ◽  
Hamilton Cycles ◽  
Szemeredi's Regularity Lemma

The notion of robust expansion has played a central role in the solution of several conjectures involving the packing of Hamilton cycles in graphs and directed graphs. These and other results usually rely on the fact that every robustly expanding (di)graph with suitably large minimum degree contains a Hamilton cycle. Previous proofs of this require Szemerédi's Regularity Lemma and so this fact can only be applied to dense, sufficiently large robust expanders. We give a proof that does not use the Regularity Lemma and, indeed, we can apply our result to sparser robustly expanding digraphs.

Antidirected Subtrees of Directed Graphs

1982 ◽  
Vol 25 (1) ◽  
pp. 119-120 ◽  
Author(s):  
Stefan A. Burr
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Personal Communication ◽  
Directed Path ◽  
Directed Tree ◽  
De Bruijn ◽  
Source Sink

The purpose of this paper is to prove the following result:Theorem. Let T be a directed tree with k arcs and with no directed path of length 2. Then if G is any directed graph with n points and at least 4kn arcs, T is a subgraph of G.It would be appropriate to call T an antidirected tree or a source-sink tree, since every point either has all its arcs directed outward or all inward. As N. G. de Bruijn has noted (personal communication), such a linear bound in n cannot hold if T is replaced by any directed graph other than a union of such trees. The above theorem strengthens one of Graham [1], where an implicit bound of c(k)n is obtained, where c(k) is exponentially large. The proof we give here is also shorter. We first give two simple lemmas. Both are essentially due to ErdÄs, but it is not clear where either first appeared. Their proofs are easy, so we give them here for completeness; in neither case do we state quite the best possible result.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

scholarly journals Counting Hamilton cycles in Dirac hypergraphs

2020 ◽  
pp. 1-23
Author(s):  
Stefan Glock ◽  
Stephen Gould ◽  
Felix Joos ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Similar Estimate ◽  
Hamilton Cycles

Abstract A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

scholarly journals Transversals in permutation groups and factorisations of complete graphs

2002 ◽  
Vol 65 (2) ◽  
pp. 277-288 ◽  
Author(s):  
Gil Kaplan ◽  
Arieh Lev
Keyword(s):  
Permutation Group ◽  
Directed Graph ◽  
Sufficient Conditions ◽  
Combinatorial Problems ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Complete Graphs ◽  
Frobenius Theorem ◽  
Disjoint Cycles ◽  
Finite Set

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

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