scholarly journals On $r$-Uniform Linear Hypergraphs with no Berge-$K_{2,t}$

10.37236/6470 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Craig Timmons
Keyword(s):  
Asymptotic Formula ◽  
Uniform Hypergraph ◽  
Linear Hypergraphs ◽  
Linear Hypergraph

Let $\mathcal{F}$ be an $r$-uniform hypergraph and $G$ be a multigraph. The hypergraph $\mathcal{F}$ is a Berge-$G$ if there is a bijection $f: E(G) \rightarrow E( \mathcal{F} )$ such that $e \subseteq f(e)$ for each $e \in E(G)$.  Given a family of multigraphs $\mathcal{G}$, a hypergraph $\mathcal{H}$ is said to be $\mathcal{G}$-free if for each $G \in \mathcal{G}$, $\mathcal{H}$ does not contain a subhypergraph that is isomorphic to a Berge-$G$. We prove bounds on the maximum number of edges in an $r$-uniform linear hypergraph that is $K_{2,t}$-free. We also determine an asymptotic formula for the maximum number of edges in a linear 3-uniform 3-partite hypergraph that is $\{C_3 , K_{2,3} \}$-free. 

scholarly journals List Coloring Hypergraphs

10.37236/401 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Penny Haxell ◽  
Jacques Verstraete
Keyword(s):  
Chromatic Number ◽  
The Other ◽  
List Coloring ◽  
Other Hand ◽  
List Chromatic Number ◽  
Linear Hypergraphs ◽  
Linear Hypergraph

Let $H$ be a hypergraph and let $L_v : v \in V(H)$ be sets; we refer to these sets as lists and their elements as colors. A list coloring of $H$ is an assignment of a color from $L_v$ to each $v \in V(H)$ in such a way that every edge of $H$ contains a pair of vertices of different colors. The hypergraph $H$ is $k$-list-colorable if it has a list coloring from any collection of lists of size $k$. The list chromatic number of $H$ is the minimum $k$ such that $H$ is $k$-list-colorable. In this paper we prove that every $d$-regular three-uniform linear hypergraph has list chromatic number at least $(\frac{\log d}{5\log \log d})^{1/2}$ provided $d$ is large enough. On the other hand there exist $d$-regular three-uniform linear hypergraphs with list chromatic number at most $\log_3 d+3$. This leaves the question open as to the existence of such hypergraphs with list chromatic number $o(\log d)$ as $d \rightarrow \infty$.

Linear Turán Numbers of Linear Cycles and Cycle-Complete Ramsey Numbers

2017 ◽  
Vol 27 (3) ◽  
pp. 358-386 ◽  
Author(s):  
CLAYTON COLLIER-CARTAINO ◽  
NATHAN GRABER ◽  
TAO JIANG
Keyword(s):  
Positive Integer ◽  
Positive Constant ◽  
Uniform Hypergraph ◽  
Ramsey Numbers ◽  
Turán Number ◽  
Formula Group ◽  
Image Position ◽  
Linear Hypergraphs

Anr-uniform hypergraph is called anr-graph. A hypergraph islinearif every two edges intersect in at most one vertex. Given a linearr-graphHand a positive integern, thelinear Turán numberexL(n,H) is the maximum number of edges in a linearr-graphGthat does not containHas a subgraph. For each ℓ ≥ 3, letCrℓdenote ther-uniform linear cycle of length ℓ, which is anr-graph with edgese1, . . .,eℓsuch that, for alli∈ [ℓ−1], |ei∩ei+1|=1, |eℓ∩e1|=1 andei∩ej= ∅ for all other pairs {i,j},i≠j. For allr≥ 3 and ℓ ≥ 3, we show that there exists a positive constantc=cr,ℓ, depending onlyrand ℓ, such that exL(n,Crℓ) ≤cn1+1/⌊ℓ/2⌋. This answers a question of Kostochka, Mubayi and Verstraëte [30]. For even ℓ, our result extends the result of Bondy and Simonovits [7] on the Turán numbers of even cycles to linear hypergraphs.Using our results on linear Turán numbers, we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constantsa=am,randb=bm,r, depending only onmandr, such that\begin{equation} R(C^r_{2m}, K^r_t)\leq a \Bigl(\frac{t}{\ln t}\Bigr)^{{m}/{(m-1)}} \quad\text{and}\quad R(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}. \end{equation}

scholarly journals On Even-Degree Subgraphs of Linear Hypergraphs

2012 ◽  
Vol 21 (1-2) ◽  
pp. 113-127 ◽  
Author(s):  
D. DELLAMONICA ◽  
P. HAXELL ◽  
T. ŁUCZAK ◽  
D. MUBAYI ◽  
B. NAGLE ◽  
...  
Keyword(s):  
Absolute Constant ◽  
Maximum Degree ◽  
Interesting Case ◽  
Uniform Hypergraph ◽  
Image Position ◽  
Linear Hypergraphs

A subgraph of a hypergraph H is even if all its degrees are positive even integers, and b-bounded if it has maximum degree at most b. Let fb(n) denote the maximum number of edges in a linearn-vertex 3-uniform hypergraph which does not contain a b-bounded even subgraph. In this paper, we show that if b ≥ 12, then for some absolute constant B, thus establishing fb(n) up to polylogarithmic factors. This leaves open the interesting case b = 2, which is the case of 2-regular subgraphs. We are able to show for some constants c, C > 0 that We conjecture that f2(n) = n1 + o(1) as n → ∞.

scholarly journals Linear Hypergraph Edge Coloring - Generalizations of the EFL Conjecture

2016 ◽  
Vol 17 ◽  
pp. 1-9
Author(s):  
Vance Faber
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Large Degree ◽  
Fano Plane ◽  
Fixed Rank ◽  
Linear Hypergraphs ◽  
Linear Hypergraph

Motivated by the Erdos-Faber-Lovász (EFL) conjecture for hypergraphs, we consider the edge coloring of linear hypergraphs. We discuss several conjectures for coloring linear hypergraphs that generalize both EFL and Vizing's theorem for graphs. For example, we conjecture that in a linear hypergraph of rank 3, the edge chromatic number is at most 2 times the maximum degree unless the hypergraph is the Fano plane where the number is 7. We show that for fixed rank sufficiently large and sufficiently large degree, the conjectures are true.

scholarly journals EIGENVALUES AND LINEAR QUASIRANDOM HYPERGRAPHS

2015 ◽  
Vol 3 ◽  
Author(s):  
JOHN LENZ ◽  
DHRUV MUBAYI
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Spectral Approach ◽  
Uniform Hypergraph ◽  
Largest Eigenvalues ◽  
Linear Hypergraph ◽  
Partial Extension

Let$p(k)$denote the partition function of$k$. For each$k\geqslant 2$, we describe a list of$p(k)-1$quasirandom properties that a$k$-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness by Kohayakawa, Rödl, and Skokan, and by Conlon, Hàn, Person, and Schacht, and the spectral approach of Friedman and Wigderson. For each of the quasirandom properties that is described, we define the largest and the second largest eigenvalues. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon, Hàn, Person, and Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung, Graham, and Wilson for graphs.

ON EDGE COLORING OF HYPERGRAPHS AND ERDÖS–FABER–LOVÁSZ CONJECTURE

2012 ◽  
Vol 04 (01) ◽  
pp. 1250003 ◽  
Author(s):  
VIJI PAUL ◽  
K. A. GERMINA
Keyword(s):  
Edge Coloring ◽  
Discrete Math ◽  
Chromatic Index ◽  
Equivalent Formulation ◽  
Lovász Conjecture ◽  
Linear Hypergraphs ◽  
Linear Hypergraph ◽  
Existing Result

The celebrated Erdös–Faber–Lovász conjecture originated in the year 1972. It can be stated as follows: any linear hypergraph on n vertices has chromatic index at most n. Different formulations of the conjecture have been obtained and the conjecture is proved to be true in some particular cases. But the problem is still unsolved in general. In this paper, we prove that the conjecture is true for all linear hypergraphs on n vertices with [Formula: see text]. It generalizes an existing result regarding an equivalent formulation of the conjecture for dense hypergraphs [A. Sanchez-Arroyo, The Erdös–Faber–Lovász conjecture for dense hypergraphs, Discrete Math.308 (2008) 991–992].

Super Edge-Connected Linear Hypergraphs

2020 ◽  
Vol 30 (03) ◽  
pp. 2040003
Author(s):  
Shangwei Lin ◽  
Jianfeng Pei ◽  
Chunfang Li
Keyword(s):  
Connected Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Edge Connectivity ◽  
Necessary And Sufficient ◽  
Linear Hypergraphs ◽  
Linear Hypergraph

A connected graph [Formula: see text] is super edge-connected, if every minimum edge-cut of [Formula: see text] is the set of edges incident with a vertex. In this paper, the concept of super edge-connectivity of graphs is generalized to hypergraphs and a necessary and sufficient condition for an [Formula: see text]-uniform and linear hypergraph with diameter at most 2 to be super edge-connected is given.

Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model

2007 ◽  
Vol 44 (02) ◽  
pp. 285-294 ◽  
Author(s):  
Qihe Tang
Keyword(s):  
Asymptotic Formula ◽  
Continuous Time ◽  
Heavy Tails ◽  
Tail Behavior ◽  
Renewal Model ◽  
Discounted Aggregate Claims ◽  
Aggregate Claims ◽  
Tail Asymptotic

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

Approximation properties of Szász type operators involving Charlier polynomials

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 479-487
Author(s):  
Didem Arı
Keyword(s):  
Asymptotic Formula ◽  
Weighted Space ◽  
Approximation Properties ◽  
Charlier Polynomials

In this paper, we give some approximation properties of Sz?sz type operators involving Charlier polynomials in the polynomial weighted space and we give the quantitative Voronovskaya-type asymptotic formula.

scholarly journals On the least common multiple of random q-integers

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Probabilistic Model ◽  
Expected Value ◽  
Random Set ◽  
Common Multiple ◽  
Least Common Multiple

AbstractFor every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.

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