scholarly journals The complexity of 2-coloring and strong coloring in uniform hypergraphs of high minimum degree

2013 ◽  
Vol Vol. 15 no. 2 (Discrete Algorithms) ◽  
Author(s):  
Edyta Szymańska
Keyword(s):  
Time Complexity ◽  
Minimum Degree ◽  
Threshold Value ◽  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Fano Plane ◽  
Uniform Hypergraphs ◽  
Discrete Algorithms ◽  
International Audience ◽  
Np Complete

Discrete Algorithms International audience In this paper we consider the problem of deciding whether a given r-uniform hypergraph H with minimum vertex degree at least c\binom|V(H)|-1r-1, or minimum degree of a pair of vertices at least c\binom|V(H)|-2r-2, has a vertex 2-coloring. Motivated by an old result of Edwards for graphs, we obtain first optimal dichotomy results for 2-colorings of r-uniform hypergraphs. For each problem, for every r≥q 3 we determine a threshold value depending on r such that the problem is NP-complete for c below the threshold, while for c strictly above the threshold it is polynomial. We provide an algorithm constructing the coloring with time complexity O(n^\lfloor 4/ε\rfloor+2\log n) with some ε>0. This algorithm becomes more efficient in the case of r=3,4,5 due to known Turán numbers of the triangle and the Fano plane. In addition, we determine the computational complexity of strong k-coloring of 3-uniform hypergraphs H with minimum vertex degree at least c\binom|V(H)|-12, for some c, leaving a gap for k≥q 5 which vanishes as k→ ∞.

scholarly journals Matchings and Hamilton cycles in hypergraphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Bipartite Graph ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Uniform Hypergraphs ◽  
International Audience

International audience It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac's theorem on Hamilton cycles for $3$-uniform hypergraphs: We say that a $3$-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. We prove that for every $\varepsilon > 0$ there is an $n_0$ such that every $3$-uniform hypergraph of order $n \geq n_0$ whose minimum degree is at least $n/4+ \varepsilon n$ contains a Hamilton cycle. Our bounds on the minimum degree are essentially best possible.

scholarly journals A linear time algorithm for finding an Euler walk in a strongly connected 3-uniform hypergraph

2012 ◽  
Vol Vol. 14 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Zbigniew Lonc ◽  
Pawel Naroski
Keyword(s):  
Linear Time ◽  
Natural Extension ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Uniform Hypergraph ◽  
Graph Theoretic ◽  
Uniform Hypergraphs ◽  
Discrete Algorithms ◽  
Strongly Connected ◽  
International Audience

Discrete Algorithms International audience By an Euler walk in a 3-uniform hypergraph H we mean an alternating sequence v(0), epsilon(1), v(1), epsilon(2), v(2), ... , v(m-1), epsilon(m), v(m) of vertices and edges in H such that each edge of H appears in this sequence exactly once and v(i-1); v(i) is an element of epsilon(i), v(i-1) not equal v(i), for every i = 1, 2, ... , m. This concept is a natural extension of the graph theoretic notion of an Euler walk to the case of 3-uniform hypergraphs. We say that a 3-uniform hypergraph H is strongly connected if it has no isolated vertices and for each two edges e and f in H there is a sequence of edges starting with e and ending with f such that each two consecutive edges in this sequence have two vertices in common. In this paper we give an algorithm that constructs an Euler walk in a strongly connected 3-uniform hypergraph (it is known that such a walk in such a hypergraph always exists). The algorithm runs in time O(m), where m is the number of edges in the input hypergraph.

scholarly journals Vertex Degree Sums for Perfect Matchings in 3-Uniform Hypergraphs

10.37236/7658 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Yi Zhang ◽  
Yi Zhao ◽  
Mei Lu
Keyword(s):  
Perfect Matching ◽  
Minimum Degree ◽  
Perfect Matchings ◽  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Degree Sum ◽  
Uniform Hypergraphs ◽  
Isolated Vertex ◽  
The One

We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-uniform hypergraph without an isolated vertex. Suppose that $H$ is a 3-uniform hypergraph whose order $n$ is sufficiently large and divisible by $3$. If $H$ contains no isolated vertex and $\deg(u)+\deg(v) > \frac{2}{3}n^2-\frac{8}{3}n+2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching. This bound is tight and the (unique) extremal hyergraph is a different space barrier from the one for the corresponding Dirac problem.

scholarly journals Vertex Degree Sums for Matchings in 3-Uniform Hypergraphs

10.37236/8627 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Yi Zhang ◽  
Yi Zhao ◽  
Mei Lu
Keyword(s):  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Degree Sum ◽  
Uniform Hypergraphs ◽  
Positive Integers ◽  
Degree Sum Condition

Let $n, s$ be positive integers such that $n$ is sufficiently large and $s\le n/3$. Suppose $H$ is a 3-uniform hypergraph of order $n$ without isolated vertices. If $\deg(u)+\deg(v) > 2(s-1)(n-1)$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a matching of size $s$. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when $s= n/3$.

scholarly journals Self-complementing permutations of k-uniform hypergraphs

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Artur Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Unique Integer ◽  
International Audience ◽  
Sigma E

Graphs and Algorithms International audience A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). <br> For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. <br> In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 <= s <= 2(l) and the following two conditions hold: <br> (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and <br> (ii) Sigma(i:lambda(vertical bar Oi vertical bar)<= l) vertical bar O-i vertical bar <= r. <br> For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.

scholarly journals Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

scholarly journals On the complexity of distributed BFS in ad hoc networks with non-spontaneous wake-ups

2013 ◽  
Vol Vol. 15 no. 3 (Distributed Computing and...) ◽  
Author(s):  
Dariusz Kowalski ◽  
Krzysztof Krzywdziński
Keyword(s):  
Ad Hoc Networks ◽  
Ad Hoc Network ◽  
Time Complexity ◽  
Ad Hoc ◽  
Threshold Value ◽  
Message Complexity ◽  
Node Network ◽  
International Audience ◽  
Hoc Networks ◽  
Point To Point

Distributed Computing and Networking International audience We study time and message complexity of the problem of building a BFS tree by a spontaneously awaken node in ad hoc network. Computation is in synchronous rounds, and messages are sent via point-to-point bi-directional links. Network topology is modeled by a graph. Each node knows only its own id and the id's of its neighbors in the network and no pre-processing is allowed; therefore the solutions to the problem of spanning a BFS tree in this setting must be distributed. We deliver a deterministic distributed solution that trades time for messages, mainly, with time complexity O(D . min(D; n=f(n)) . logD . log n) and with the number of point-to-point messages sent O(n. (min(D; n=f(n))+f(n)) . logD. log n), for any n-node network with diameter D and for any monotonically non-decreasing sub-linear integer function f. Function f in the above formulas come from the threshold value on node degrees used by our algorithms, in the sense that nodes with degree at most f(n) are treated differently that the other nodes. This yields the first BFS-finding deterministic distributed algorithm in ad hoc networks working in time o(n) and with o(n2) message complexity, for some suitable functions f(n) = o(n= log2 n), provided D = o(n= log4 n).

scholarly journals A note on contracting claw-free graphs

2013 ◽  
Vol Vol. 15 no. 2 (Discrete Algorithms) ◽  
Author(s):  
Jiří Fiala ◽  
Marcin Kamiński ◽  
Daniël Paulusma
Keyword(s):  
Line Graphs ◽  
Natural Question ◽  
Fixed Target ◽  
Induced Subgraphs ◽  
Graph Operations ◽  
Spanning Subgraphs ◽  
Discrete Algorithms ◽  
International Audience ◽  
Np Complete ◽  
Host Graph

Discrete Algorithms International audience A graph containment problem is to decide whether one graph called the host graph can be modified into some other graph called the target graph by using a number of specified graph operations. We consider edge deletions, edge contractions, vertex deletions and vertex dissolutions as possible graph operations permitted. By allowing any combination of these four operations we capture the following problems: testing on (induced) minors, (induced) topological minors, (induced) subgraphs, (induced) spanning subgraphs, dissolutions and contractions. We show that these problems stay NP-complete even when the host and target belong to the class of line graphs, which form a subclass of the class of claw-free graphs, i.e., graphs with no induced 4-vertex star. A natural question is to study the computational complexity of these problems if the target graph is assumed to be fixed. We show that these problems may become computationally easier when the host graphs are restricted to be claw-free. In particular we consider the problems that are to test whether a given host graph contains a fixed target graph as a contraction.

Minimum Degrees and Codegrees of Ramsey-Minimal 3-Uniform Hypergraphs

2016 ◽  
Vol 25 (6) ◽  
pp. 850-869
Author(s):  
DENNIS CLEMENS ◽  
YURY PERSON
Keyword(s):  
Vertex Degree ◽  
Uniform Hypergraph ◽  
Minimal Graphs ◽  
Uniform Hypergraphs ◽  
Minimum Degrees ◽  
Monochromatic Copy

A uniform hypergraph H is called k-Ramsey for a hypergraph F if, no matter how one colours the edges of H with k colours, there is always a monochromatic copy of F. We say that H is k-Ramsey-minimal for F if H is k-Ramsey for F but every proper subhypergraph of H is not. Burr, Erdős and Lovasz studied various parameters of Ramsey-minimal graphs. In this paper we initiate the study of minimum degrees and codegrees of Ramsey-minimal 3-uniform hypergraphs. We show that the smallest minimum vertex degree over all k-Ramsey-minimal 3-uniform hypergraphs for Kt(3) is exponential in some polynomial in k and t. We also study the smallest possible minimum codegree over 2-Ramsey-minimal 3-uniform hypergraphs.

scholarly journals Weak and Strong Versions of the 1-2-3 Conjecture for Uniform Hypergraphs

10.37236/5709 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Alan Frieze ◽  
Laars Helenius
Keyword(s):  
Weight Function ◽  
Lower Bounds ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs ◽  
Np Complete ◽  
Almost All

Given an $r$-uniform hypergraph $H=(V,E)$ and a weight function $\omega:E\to\{1,\dots,w\}$, a coloring of vertices of~$H$, induced by~$\omega$, is defined by $c(v) = \sum_{e\ni v} w(e)$ for all $v\in V$. If there exists such a coloring that is strong (that means in each edge no color appears more than once), then we say that $H$ is strongly $w$-weighted. Similarly, if the coloring is weak (that means there is no monochromatic edge), then we say that $H$ is weakly $w$-weighted. In this paper, we show that almost all 3 or 4-uniform hypergraphs are strongly 2-weighted (but not 1-weighted) and almost all $5$-uniform hypergraphs are either 1 or 2 strongly weighted (with a nontrivial distribution). Furthermore, for $r\ge 6$ we show that almost all $r$-uniform hypergraphs are strongly 1-weighted. We complement these results by showing that almost all 3-uniform hypergraphs are weakly 2-weighted but not 1-weighted and for $r\ge 4$ almost all $r$-uniform hypergraphs are weakly 1-weighted. These results extend a previous work of Addario-Berry, Dalal and Reed for graphs. We also prove general lower bounds and show that there are $r$-uniform hypergraphs which are not strongly $(r^2-r)$-weighted and not weakly 2-weighted. Finally, we show that determining whether a particular uniform hypergraph is strongly 2-weighted is NP-complete.

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