scholarly journals The Rupture Degree of <i>k</i>-Uniform Linear Hypergraph

2021 ◽  
Vol 12 (07) ◽  
pp. 556-562
Author(s):  
Ning Zhao
Keyword(s):  
Linear Hypergraph

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 $d \rightarrow \infty$.

scholarly journals Spectral Extremal Results for Hypergraphs

10.37236/9018 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Yuan Hou ◽  
An Chang ◽  
Joshua Cooper
Keyword(s):  
Structural Properties ◽  
Spectral Radius ◽  
Spectral Methods ◽  
Adjacency Tensor ◽  
Uniform Hypergraphs ◽  
Linear Hypergraph

Let $F$ be a graph. A hypergraph is called Berge $F$ if it can be obtained by replacing each edge in $F$ by a hyperedge containing it. Given a family of graphs $\mathcal{F}$, we say that a hypergraph $H$ is Berge $\mathcal{F}$-free if for every $F \in \mathcal{F}$, the hypergraph $H$ does not contain a Berge $F$ as a subhypergraph. In this paper we investigate the connections between spectral radius of the adjacency tensor and structural properties of a linear hypergraph. In particular, we obtain a spectral version of Turán-type problems over linear $k$-uniform hypergraphs by using spectral methods, including a tight result on Berge $C_4$-free linear $3$-uniform hypergraphs.

The Vulnerability of k-Uniform Linear Hypergraph

2021 ◽  
pp. 2142003
Author(s):  
Ning Zhao ◽  
Yinkui Li
Keyword(s):  
Connected Components ◽  
Linear Hypergraph

For a given graph [Formula: see text], denote by [Formula: see text] and [Formula: see text] the order of the largest component and the number of connected components of [Formula: see text], respectively. The scattering number of [Formula: see text] is defined as [Formula: see text]. The tenacity of [Formula: see text] is defined as [Formula: see text]. These two theoretical parameters are important combinatorial parameters for measuring the vulnerability of networks. In this paper, we determine the scattering number and tenacity of [Formula: see text]-uniform linear hypergraph [Formula: see text].

Configurations containing a given linear hypergraph

2017 ◽  
Vol 87 (3) ◽  
pp. 356-361
Author(s):  
Peter J. Dukes ◽  
Garret Flowers
Keyword(s):  
Linear Hypergraph

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.

On hypergraph coloring and 3-uniform linear hypergraph set-indexers of a graph

2015 ◽  
Vol 07 (02) ◽  
pp. 1550015
Author(s):  
Viji Paul ◽  
K A Germina
Keyword(s):  
Linear Set ◽  
Hypergraph Coloring ◽  
Linear Hypergraph

For a graph G = (V, E) and a nonempty set X, a linear hypergraph set-indexer (LHSI) is a function f : V(G) → 2X satisfying the following conditions: (i) f is injective (ii) the ordered pair Hf(G) = (X, f(V)), where f(V) = {f(v) : v ∈ V(G)}, is a linear hypergraph (iii) the induced function f⊕ : E → 2X defined by f⊕(uv) = f(u) ⊕ f(v), for all uv ∈ E is injective and (iv) Hf⊕(G) = (X, f⊕(E)), where f⊕(E) = {f⊕(e) : e ∈ E}, is a linear hypergraph. It is shown that a 3-uniform LHSI of a graph, corresponding to upper LHSI number, is unique up to isomorphism and possible relations between the coloring numbers of a given graph and the two linear set-indexing hypergraphs are established. Also, we show that the two hypergraphs associated with an extremal 3-uniform LHSI of a graph are 2-colorable and the corresponding induced edge hypergraphs are self-dual.

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