scholarly journals Edge Balanced 3-Uniform Hypergraph Designs

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1353
Author(s):  
Paola Bonacini ◽  
Mario Gionfriddo ◽  
Lucia Marino
Keyword(s):  
Chromatic Number ◽  
Uniform Hypergraph ◽  
Strong Chromatic Number

In this paper, we completely determine the spectrum of edge balanced H-designs, where H is a 3-uniform hypergraph with 2 or 3 edges, such that H has strong chromatic number χs(H)=3.

On the strong chromatic number of a random 3-uniform hypergraph

2021 ◽  
Vol 344 (3) ◽  
pp. 112231
Author(s):  
Arseniy E. Balobanov ◽  
Dmitry A. Shabanov
Keyword(s):  
Chromatic Number ◽  
Uniform Hypergraph ◽  
Strong Chromatic Number

List Colourings of Regular Hypergraphs

2012 ◽  
Vol 21 (1-2) ◽  
pp. 315-322 ◽  
Author(s):  
DAVID SAXTON ◽  
ANDREW THOMASON
Keyword(s):  
Chromatic Number ◽  
Uniform Hypergraph ◽  
List Chromatic Number

We show that the list chromatic number of a simpled-regularr-uniform hypergraph is at least (1/2rlog(2r2) +o(1)) logdifdis large.

Maximal antiramsey graphs and the strong chromatic number

1989 ◽  
Vol 13 (3) ◽  
pp. 263-282 ◽  
Author(s):  
S. A. Burr ◽  
P. Erdös ◽  
R. L. Graham ◽  
V. T. Sós
Keyword(s):  
Chromatic Number ◽  
Strong Chromatic Number

On strict strong coloring of graphs

2020 ◽  
pp. 2150040
Author(s):  
A. Mohammed Abid ◽  
T. R. Ramesh Rao
Keyword(s):  
Chromatic Number ◽  
Proper Coloring ◽  
Color Class ◽  
Minimum Number ◽  
Strong Chromatic Number

A strict strong coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which every vertex of the graph is adjacent to every vertex of some color class. The minimum number of colors required for a strict strong coloring of [Formula: see text] is called the strict strong chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we characterize the results on strict strong coloring of Mycielskian graphs and iterated Mycielskian graphs.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals A Note on Embedding Hypertrees

10.37236/256 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Po-Shen Loh
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Classical Result ◽  
Uniform Hypergraph ◽  
Uniform Hypergraphs

A classical result from graph theory is that every graph with chromatic number $\chi > t$ contains a subgraph with all degrees at least $t$, and therefore contains a copy of every $t$-edge tree. Bohman, Frieze, and Mubayi recently posed this problem for $r$-uniform hypergraphs. An $r$-tree is a connected $r$-uniform hypergraph with no pair of edges intersecting in more than one vertex, and no sequence of distinct vertices and edges $(v_1, e_1, \ldots, v_k, e_k)$ with all $e_i \ni \{v_i, v_{i+1}\}$, where we take $v_{k+1}$ to be $v_1$. Bohman, Frieze, and Mubayi proved that $\chi > 2rt$ is sufficient to embed every $r$-tree with $t$ edges, and asked whether the dependence on $r$ was necessary. In this note, we completely solve their problem, proving the tight result that $\chi > t$ is sufficient to embed any $r$-tree with $t$ edges.

scholarly journals Colorful Subhypergraphs in Kneser Hypergraphs

10.37236/3573 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Frédéric Meunier
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Bipartite Graphs ◽  
Uniform Hypergraph ◽  
Proper Coloring ◽  
Kneser Graph ◽  
Complete Bipartite Graphs ◽  
Definition Of ◽  
Ky Fan

Using a $\mathbb{Z}_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a uniform hypergraph).

scholarly journals An asymptotic bound for the strong chromatic number

2019 ◽  
Vol 28 (5) ◽  
pp. 768-776
Author(s):  
Allan Lo ◽  
Nicolás Sanhueza-Matamala
Keyword(s):  
Chromatic Number ◽  
Asymptotic Bound ◽  
Vertex Set ◽  
Strong Chromatic Number ◽  
Proper Vertex ◽  
Vertex Colouring

AbstractThe strong chromatic number χs(G) of a graph G on n vertices is the least number r with the following property: after adding $r\lceil n/r\rceil-n$ isolated vertices to G and taking the union with any collection of spanning disjoint copies of Kr in the same vertex set, the resulting graph has a proper vertex colouring with r colours. We show that for every c > 0 and every graph G on n vertices with Δ(G) ≥ cn, χs(G) ≤ (2+o(1))Δ(G), which is asymptotically best possible.

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