On chromatic number of finite set-systems

1968 ◽  
Vol 19 (1-2) ◽  
pp. 59-67 ◽  
Author(s):  
L. Lovász
Keyword(s):  
Chromatic Number ◽  
Set Systems ◽  
Finite Set

scholarly journals Strong T-Coloring of Graphs

2019 ◽  
Vol 8 (12) ◽  
pp. 4677-4681
Keyword(s):  
Chromatic Number ◽  
Maximum Value ◽  
Finite Set ◽  
Minimum Number ◽  
Edge Span ◽  
Positive Integers

A 𝑻-coloring of a graph 𝑮 = (𝑽,𝑬) is the generalized coloring of a graph. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set T of positive integers containing 𝟎 , a 𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻. We define Strong 𝑻-coloring (S𝑻-coloring , in short), as a generalization of 𝑻-coloring as follows. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set 𝑻 of positive integers containing 𝟎, a S𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻 and |𝒇(𝒖) − 𝒇(𝒘)| ≠ |𝒇(𝒙) − 𝒇(𝒚)| for any two distinct edges 𝒖𝒘, 𝒙𝒚 in 𝑬(𝑮). The S𝑻-Chromatic number of 𝑮 is the minimum number of colors needed for a S𝑻-coloring of 𝑮 and it is denoted by 𝝌𝑺𝑻(𝑮) . For a S𝑻 coloring 𝒄 of a graph 𝑮 we define the 𝒄𝑺𝑻- span 𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all pairs 𝒖, 𝒗 of vertices of 𝑮 and the S𝑻 -span 𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒔𝒑𝑺𝑻(𝑮) = min 𝒔𝒑𝑺𝑻 𝒄 (𝑮) where the minimum is taken over all ST-coloring c of G. Similarly the 𝒄𝑺𝑻-edgespan 𝒆𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all edges 𝒖𝒗 of 𝑮 and the S𝑻-edge span 𝒆𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒆𝒔𝒑𝑺𝑻(𝑮) = min 𝒆𝒔𝒑𝑺𝑻 𝒄 𝑮 where the minimum is taken over all ST-coloring c of G. In this paper we discuss these concepts namely, S𝑻- chromatic number, 𝒔𝒑𝑺𝑻(𝑮) , and 𝒆𝒔𝒑𝑺𝑻(𝑮) of graphs.

Colour Classes for r-Graphs

1972 ◽  
Vol 15 (3) ◽  
pp. 349-354 ◽  
Author(s):  
E. J. Cockayne
Keyword(s):  
Structural Properties ◽  
Chromatic Number ◽  
Ramsey Numbers ◽  
Graph Theoretic ◽  
Special Cases ◽  
Finite Set ◽  
Colour Classes

By an r-graph G we mean a finite set V(G) of elements called vertices and a set E(G) of some of the r-subsets of V(G) called edges. This paper defines certain colour classes of r-graphs which connect the material of a variety of recent graph theoretic literature in that many existing results may be reformulated as structural properties of the classes for some special cases of r-graphs. It is shown that the concepts of Ramsey Numbers, chromatic number and index may be defined in terms of these classes. These concepts and some of their properties are generalized. The final subsection compares two existing bounds for the chromatic number of a graph.

scholarly journals Distance graphs with maximum chromatic number

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Javier Barajas ◽  
Oriol Serra
Keyword(s):  
Chromatic Number ◽  
Distance Graph ◽  
Distance Graphs ◽  
Maximum Value ◽  
Finite Set ◽  
International Audience

International audience Let $D$ be a finite set of integers. The distance graph $G(D)$ has the set of integers as vertices and two vertices at distance $d ∈D$ are adjacent in $G(D)$. A conjecture of Xuding Zhu states that if the chromatic number of $G (D)$ achieves its maximum value $|D|+1$ then the graph has a clique of order $|D|$. We prove that the chromatic number of a distance graph with $D=\{ a,b,c,d\}$ is five if and only if either $D=\{1,2,3,4k\}$ or $D=\{ a,b,a+b,a+2b\}$ with $a \equiv 0 (mod 2)$ and $b \equiv 1 (mod 2)$. This confirms Zhu's conjecture for $|D|=4$.

scholarly journals INFINITE COMBINATORICS PLAIN AND SIMPLE

2018 ◽  
Vol 83 (3) ◽  
pp. 1247-1281 ◽  
Author(s):  
DÁNIEL T. SOUKUP ◽  
LAJOS SOUKUP
Keyword(s):  
Chromatic Number ◽  
Set Theory ◽  
Wide Applicability ◽  
Elementary Submodels ◽  
Set Systems ◽  
Point Set ◽  
Infinite Combinatorics ◽  
Sparse Set ◽  
Paradoxical Decompositions ◽  
General Method

AbstractWe explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

On chromatic number of graphs and set-systems

1966 ◽  
Vol 17 (1-2) ◽  
pp. 61-99 ◽  
Author(s):  
P. Erdős ◽  
A. Hajnal
Keyword(s):  
Chromatic Number ◽  
Set Systems

scholarly journals Set Systems with Finite Chromatic Number

1989 ◽  
Vol 10 (6) ◽  
pp. 543-549 ◽  
Author(s):  
Péter Komjáth
Keyword(s):  
Chromatic Number ◽  
Set Systems

scholarly journals On the chromatic number of set systems

2001 ◽  
Vol 19 (2) ◽  
pp. 87-98 ◽  
Author(s):  
Alexandr Kostochka ◽  
Dhruv Mubayi ◽  
Vojtěch Rödl ◽  
Prasad Tetali
Keyword(s):  
Chromatic Number ◽  
Set Systems
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