A Dirac-type Theorem for Berge Cycles in Random Hypergraphs

A Hamilton Berge cycle of a hypergraph on $n$ vertices is an alternating sequence $(v_1, e_1, v_2, \ldots, v_n, e_n)$ of distinct vertices $v_1, \ldots, v_n$ and distinct hyperedges $e_1, \ldots, e_n$ such that $\{v_1,v_n\}\subseteq e_n$ and $\{v_i, v_{i+1}\} \subseteq e_i$ for every $i\in [n-1]$. We prove the following Dirac-type theorem about Berge cycles in the binomial random $r$-uniform hypergraph $H^{(r)}(n,p)$: for every integer $r \geq 3$, every real $\gamma>0$ and $p \geq \frac{\ln^{17r} n}{n^{r-1}}$ asymptotically almost surely, every spanning subgraph $H \subseteq H^{(r)}(n,p)$ with minimum vertex degree $\delta_1(H) \geq \left(\frac{1}{2^{r-1}} + \gamma\right) p \binom{n}{r-1}$ contains a Hamilton Berge cycle. The minimum degree condition is asymptotically tight and the bound on $p$ is optimal up to some polylogarithmic factor.

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A Dirac-type theorem for Hamilton Berge cycles in random hypergraphs

Electronic Notes in Discrete Mathematics ◽

10.1016/j.endm.2016.09.032 ◽

2016 ◽

Vol 54 ◽

pp. 181-186 ◽

Cited By ~ 4

Author(s):

Dennis Clemens ◽

Julia Ehrenmüller ◽

Yury Person

Keyword(s):

Type Theorem ◽

Dirac Type ◽

Random Hypergraphs

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Vertex Degree Sums for Perfect Matchings in 3-Uniform Hypergraphs

The Electronic Journal of Combinatorics ◽

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.

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Large Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition

The Electronic Journal of Combinatorics ◽

10.37236/7049 ◽

2017 ◽

Vol 24 (3) ◽

Cited By ~ 2

Author(s):

András Gyárfás ◽

Gábor Sárközy

Keyword(s):

Complete Graph ◽

Minimum Degree ◽

Complete Graphs ◽

Connected Component ◽

Degree Condition ◽

Colored Graphs ◽

Spanning Subgraph ◽

Delta G

It is well-known that in every $k$-coloring of the edges of the complete graph $K_n$ there is a monochromatic connected component of order at least ${n\over k-1}$. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For $k=2$ the authors proved that $\delta(G)\ge{3n\over 4}$ ensures a monochromatic connected component with at least $\delta(G)+1$ vertices in every $2$-coloring of the edges of a graph $G$ with $n$ vertices. This result is sharp, thus for $k=2$ we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is that for larger values of $k$ the situation is different, graphs of minimum degree $(1-\epsilon_k)n$ can replace complete graphs and still there is a monochromatic connected component of order at least ${n\over k-1}$, in fact $$\delta(G)\ge \left(1 - \frac{1}{1000(k-1)^9}\right)n$$ suffices.Our second result is an improvement of this bound for $k=3$. If the edges of $G$ with $\delta(G)\geq {9n\over 10}$ are $3$-colored, then there is a monochromatic component of order at least ${n\over 2}$. We conjecture that this can be improved to ${7n\over 9}$ and for general $k$ we conjecture the following: if $k\geq 3$ and $G$ is a graph of order $n$ such that $\delta(G)\geq \left( 1 - \frac{k-1}{k^2}\right)n$, then in any $k$-coloring of the edges of $G$ there is a monochromatic connected component of order at least ${n\over k-1}$.

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

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.596 ◽

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→ ∞.

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Some degree conditions for P≥k-factor covered graphs

RAIRO - Operations Research ◽

10.1051/ro/2021140 ◽

2021 ◽

Author(s):

Guowei Dai ◽

Zan-Bo Zhang ◽

Yicheng Hang ◽

Xiaoyan Zhang

Keyword(s):

Planar Graph ◽

Maximum Degree ◽

Minimum Degree ◽

Discrete Mathematics ◽

Sufficient Condition ◽

Degree Condition ◽

Spanning Subgraph ◽

K Factor ◽

The Relationship ◽

Path Factor

A spanning subgraph of a graph $G$ is called a path-factor of $G$ if its each component is a path. A path-factor is called a $\mathcal{P}_{\geq k}$-factor of $G$ if its each component admits at least $k$ vertices, where $k\geq2$. Zhang and Zhou [\emph{Discrete Mathematics}, \textbf{309}, 2067-2076 (2009)] defined the concept of $\mathcal{P}_{\geq k}$-factor covered graphs, i.e., $G$ is called a $\mathcal{P}_{\geq k}$-factor covered graph if it has a $\mathcal{P}_{\geq k}$-factor covering $e$ for any $e\in E(G)$. In this paper, we firstly obtain a minimum degree condition for a planar graph being a $\mathcal{P}_{\geq 2}$-factor and $\mathcal{P}_{\geq 3}$-factor covered graph, respectively. Secondly, we investigate the relationship between the maximum degree of any pairs of non-adjacent vertices and $\mathcal{P}_{\geq k}$-factor covered graphs, and obtain a sufficient condition for the existence of $\mathcal{P}_{\geq2}$-factor and $\mathcal{P}_{\geq 3}$-factor covered graphs, respectively.

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The Multiple-Orientability Thresholds for Random Hypergraphs

Combinatorics Probability Computing ◽

10.1017/s0963548315000334 ◽

2015 ◽

Vol 25 (6) ◽

pp. 870-908 ◽

Cited By ~ 1

Author(s):

NIKOLAOS FOUNTOULAKIS ◽

MEGHA KHOSLA ◽

KONSTANTINOS PANAGIOTOU

Keyword(s):

Load Balancing ◽

Hard Disk ◽

Maximum Load ◽

Disk Arrays ◽

Uniform Hypergraph ◽

Cuckoo Hashing ◽

Uniform Hypergraphs ◽

Random Hypergraphs ◽

Do So ◽

Parallel Access

Ak-uniform hypergraphH= (V, E) is called ℓ-orientable if there is an assignment of each edgee∈Eto one of its verticesv∈esuch that no vertex is assigned more than ℓ edges. LetHn,m,kbe a hypergraph, drawn uniformly at random from the set of allk-uniform hypergraphs withnvertices andmedges. In this paper we establish the threshold for the ℓ-orientability ofHn,m,kfor allk⩾ 3 and ℓ ⩾ 2, that is, we determine a critical quantityc*k,ℓsuch that with probability 1 −o(1) the graphHn,cn,khas an ℓ-orientation ifc<c*k,ℓ, but fails to do so ifc>c*k,ℓ.Our result has various applications, including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

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A New Bound on the Domination Number of Graphs with Minimum Degree Two

The Electronic Journal of Combinatorics ◽

10.37236/499 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 1

Author(s):

Michael A. Henning ◽

Ingo Schiermeyer ◽

Anders Yeo

Keyword(s):

Connected Graph ◽

Minimum Degree ◽

Domination Number ◽

Degree Condition ◽

Cut Vertex ◽

Delta G ◽

Vertex Disjoint ◽

Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

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On Komlós’ tiling theorem in random graphs

Combinatorics Probability Computing ◽

10.1017/s0963548319000129 ◽

2019 ◽

Vol 29 (1) ◽

pp. 113-127

Author(s):

Rajko Nenadov ◽

Nemanja Škorić

Keyword(s):

Chromatic Number ◽

Random Graph ◽

Random Graphs ◽

Upper Bound ◽

Minimum Degree ◽

Arbitrary Graph ◽

Spanning Subgraph ◽

Delta G ◽

Image Position ◽

Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

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