scholarly journals A Note on the Linear Cycle Cover Conjecture of Gyárfás and Sárközy

10.37236/7329 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Beka Ergemlidze ◽  
Ervin Győri ◽  
Abhishek Methuku
Keyword(s):  
Independent Set ◽  
Uniform Hypergraph ◽  
Edge Disjoint ◽  
Cycle Cover ◽  
Vertex Set ◽  
Cyclic Sequence

A linear cycle in a $3$-uniform hypergraph $H$ is a cyclic sequence of hyperedges such that any two consecutive hyperedges intersect in exactly one element and non-consecutive hyperedges are disjoint. Let $\alpha(H)$ denote the size of a largest independent set of $H$.We show that the vertex set of every $3$-uniform hypergraph $H$ can be covered by at most $\alpha(H)$ edge-disjoint linear cycles (where we accept a vertex and a hyperedge as a linear cycle), proving a weaker version of a conjecture of Gyárfás and Sárközy.

scholarly journals Packing Loose Hamilton Cycles

2017 ◽  
Vol 26 (6) ◽  
pp. 839-849
Author(s):  
ASAF FERBER ◽  
KYLE LUH ◽  
DANIEL MONTEALEGRE ◽  
OANH NGUYEN
Keyword(s):  
Packing Problem ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Threshold Probability ◽  
Edge Disjoint ◽  
Formula Group ◽  
Vertex Set ◽  
Substantial Progress ◽  
Image Position

A subsetCof edges in ak-uniform hypergraphHis aloose Hamilton cycleifCcovers all the vertices ofHand there exists a cyclic ordering of these vertices such that the edges inCare segments of that order and such that every two consecutive edges share exactly one vertex. The binomial randomk-uniform hypergraphHkn,phas vertex set [n] and an edge setEobtained by adding eachk-tuplee∈ ($\binom{[n]}{k}$) toEwith probabilityp, independently at random.Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all buto(|E|) edges, referred to as thepacking problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle inHkn,pis$p=\Theta\biggl(\frac{\log n}{n^{k-1}}\biggr),$the best known bounds for the packing problem are aroundp= polylog(n)/n. Here we make substantial progress and prove the following asymptotically (up to a polylog(n) factor) best possible result: forp≥ logCn/nk−1, a randomk-uniform hypergraphHkn,pwith high probability contains$N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$edge-disjoint loose Hamilton cycles.Our proof utilizes and modifies the idea of ‘online sprinkling’ recently introduced by Vu and the first author.

scholarly journals On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽  
2021 ◽  
Author(s):  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Polynomial Time ◽  
Disjoint Paths ◽  
Structural Parameter ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithms ◽  
Edge Disjoint ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

The Generalized Connectivity of Generalized Petersen Graph

2021 ◽  
Author(s):  
Yuan Si ◽  
Ping Li ◽  
Yuzhi Xiao ◽  
Jinxia Liang
Keyword(s):  
Steiner Trees ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Edge Disjoint ◽  
Generalized Connectivity ◽  
Vertex Set

For a vertex set [Formula: see text] of [Formula: see text], we use [Formula: see text] to denote the maximum number of edge-disjoint Steiner trees of [Formula: see text] such that any two of such trees intersect in [Formula: see text]. The generalized [Formula: see text]-connectivity of [Formula: see text] is defined as [Formula: see text]. We get that for any generalized Petersen graph [Formula: see text] with [Formula: see text], [Formula: see text] when [Formula: see text]. We give the values of [Formula: see text] for Petersen graph [Formula: see text], where [Formula: see text], and the values of [Formula: see text] for generalized Petersen graph [Formula: see text], where [Formula: see text] and [Formula: see text].

On complete subgraphs of different orders

1976 ◽  
Vol 79 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Intersection Graph ◽  
Edge Disjoint ◽  
Vertex Set

Let S be a set and let {X1, …, Xn} = be a family of distinct subsets of S. The intersection graph Ω() of has vertex set {X1, …, Xn} and XiXj (i ≠ j) is an edge of Ω() if and only if Xi ∩ Xi ≠ Ø (c.f. (6)). It is easily seen, (7), that every graph is an intersection graph, in other words every graph can be represented by subsets ofa set. Moreover, it was proved by Erdös, Goodman and Pósa (5) that if a graph has n ≥ 4 vertices then one can find a representing set with at most [n2/4] elements. This assertion is an immediate consequence of the result, (5), that the edges of a graph with n ≥ 1 vertices can be covered with at most [n2/4] edge disjoint triangles and edges. We say that a graph G is covered with the subgraphs G1, …, Gk, if every edge of G is in at least one Gi. One of the aims of this note is to prove an extension of this result, pro-posed by Erdös (4).

scholarly journals Edge-Disjoint Induced Subgraphs with Given Minimum Degree

10.37236/2882 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Independent Set ◽  
Induced Subgraphs ◽  
Asymptotically Optimal ◽  
Edge Disjoint

Let $h$ be a given positive integer. For a graph with $n$ vertices and $m$ edges, what is the maximum number of pairwise edge-disjoint {\em induced} subgraphs, each having  minimum degree at least $h$? There are examples for which this number is $O(m^2/n^2)$. We prove that this bound is achievable for all graphs with polynomially many edges. For all $\epsilon > 0$, if $m \ge n^{1+\epsilon}$, then there are always $\Omega(m^2/n^2)$ pairwise edge-disjoint induced subgraphs, each having  minimum degree at least $h$. Furthermore, any two subgraphs intersect in an independent set of size at most $1+ O(n^3/m^2)$, which is shown to be asymptotically optimal.

scholarly journals Monochromatic Path and Cycle Partitions in Hypergraphs

10.37236/2631 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
András Gyárfás ◽  
Gábor N. Sárközy
Keyword(s):  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Vertex Set

Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs.We show that $r$-colored $r$-uniform complete hypergraphs can be partitioned into monochromatic Berge-paths of distinct colors. Also, apart from $2k-5$ vertices, $2$-colored $k$-uniform hypergraphs can be partitioned into two monochromatic loose paths.In general, we prove that in any $r$-coloring of a $k$-uniform hypergraph there is a partition of the vertex set intomonochromatic loose cycles such that their number depends only on $r$ and $k$.

scholarly journals The Bounds of the Edge Number in Generalized Hypertrees

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 2 ◽  
Author(s):  
Ke Zhang ◽  
Haixing Zhao ◽  
Zhonglin Ye ◽  
Yu Zhu ◽  
Liang Wei
Keyword(s):  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Number Of Components ◽  
Uniform Hypergraphs ◽  
Vertex Set

A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T − E is a fixed value k   ( 2 ≤ k ≤ r ) . We focus on the case that G H T − E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n − r + 1 edges if r ≥ 3   and   n ≥ 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T − E has exactly k   ( 2 ≤ k ≤ r − 1 ) components.

scholarly journals The Minimum Number of Edges in Uniform Hypergraphs with Property O

2017 ◽  
Vol 27 (4) ◽  
pp. 531-538 ◽  
Author(s):  
DWIGHT DUFFUS ◽  
BILL KAY ◽  
VOJTĚCH RÖDL
Keyword(s):  
Linear Order ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Minimum Number ◽  
Vertex Set

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if, for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.

scholarly journals Hypergraph Independent Sets

2012 ◽  
Vol 22 (1) ◽  
pp. 9-20 ◽  
Author(s):  
JONATHAN CUTLER ◽  
A. J. RADCLIFFE
Keyword(s):  
Upper Bound ◽  
Independent Set ◽  
Independent Sets ◽  
Extremal Problems ◽  
Regularity Lemma ◽  
Uniform Hypergraph ◽  
Fixed Size

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices isj-independentif its intersection with any edge has size strictly less thanj. The Kruskal–Katona theorem implies that in anr-uniform hypergraph with a fixed size and order, the hypergraph with the mostr-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number ofj-independent sets in anr-uniform hypergraph.

scholarly journals Thek-Zero-Divisor Hypergraph of a Commutative Ring

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Ch. Eslahchi ◽  
A. M. Rahimi
Keyword(s):  
Commutative Ring ◽  
Nilpotent Element ◽  
Zero Divisor ◽  
Clique Number ◽  
Common Point ◽  
Prime Ideals ◽  
Uniform Hypergraph ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and thek-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. LetRbe a commutative ring andkan integer strictly larger than2. Ak-uniform hypergraphHk(R)with the vertex setZ(R,k), the set of allk-zero-divisors inR, is associated toR, where eachk-subset ofZ(R,k)that satisfies thek-zero-divisor condition is an edge inHk(R). It is shown that ifRhas two prime idealsP1andP2with zero their only common point, thenHk(R)is a bipartite (2-colorable) hypergraph with partition setsP1−Z′andP2−Z′, whereZ′is the set of all zero divisors ofRwhich are notk-zero-divisors inR. IfRhas a nonzero nilpotent element, then a lower bound for the clique number ofH3(R)is found. Also, we have shown thatH3(R)is connected with diameter at most 4 wheneverx2≠0for all3-zero-divisorsxofR. Finally, it is shown that for any finite nonlocal ringR, the hypergraphH3(R)is complete if and only ifRis isomorphic toZ2×Z2×Z2.

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