scholarly journals Sum List Coloring $2 \times n$ Arrays

10.37236/1669 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Garth Isaak
Keyword(s):  
Cartesian Product ◽  
Edge Coloring ◽  
Vertex Coloring ◽  
List Coloring ◽  
Proper Coloring ◽  
List Edge Coloring ◽  
Choice Number ◽  
Sum Choice Number

A graph is $f$-choosable if for every collection of lists with list sizes specified by $f$ there is a proper coloring using colors from the lists. The sum choice number is the minimum over all choosable functions $f$ of the sum of the sizes in $f$. We show that the sum choice number of a $2 \times n$ array (equivalent to list edge coloring $K_{2,n}$ and to list vertex coloring the cartesian product $K_2 \square K_n$) is $n^2 + \lceil 5n/3 \rceil$.

Graph Coloring

2013 ◽  
pp. 106-126
Author(s):  
Faraz Dadgostari ◽  
Mahtab Hosseininia
Keyword(s):  
Graph Coloring ◽  
Edge Coloring ◽  
Graph Labeling ◽  
Vertex Coloring ◽  
List Coloring ◽  
Simple Type ◽  
Critical Graphs ◽  
Classification Of Graphs ◽  
Edge List

In this chapter a particular type of graph labeling, called graph coloring, is introduced and discussed. In the first part, the simple type of coloring, vertex coloring, is focused. Thus, concerning vertex coloring, some terms and definitions are introduced. Next, some theorems and applying those theorems, some coloring algorithms and applications are introduced. At last, some helpful concepts such as critical graphs, list coloring, and vertex decomposition are presented and discussed. In the second section, edge coloring is focused. Thus, concerning edge coloring, some terms and definitions are described, some important information about edge chromatic number and edge list coloring is presented, and applying them, classification of graphs using the coloring approach is summarized. At last some helpful concepts such as edge list coloring and edge decomposition are illustrated and discussed.

More on Single Valued Neutrosophic R-dynamic Vertex Coloring and R-dynamic Edge Coloring of graphs

2021 ◽  
pp. 16-27
Author(s):  
Aparna V. ◽  
◽  
Mohanapriya N. ◽  
Broumi Said ◽  
◽  
...  
Keyword(s):  
Cartesian Product ◽  
Edge Coloring ◽  
Vertex Coloring ◽  
Neutrosophic Sets

The notion of neutrosophic sets facilitates the analysis of values that are unclear or indeterminate. In this paper, we discuss the single-valued neutrosophic R-dynamic vertex coloring of the Cartesian product of SVNG’sand join of SVG's. Further, we have described the concept of single-valued neutrosophic R-dynamic edge coloring and provided some examples and theorems.

scholarly journals Sum choice number of generalizedθ-graphs

2017 ◽  
Vol 340 (11) ◽  
pp. 2633-2640 ◽  
Author(s):  
Christoph Brause ◽  
Arnfried Kemnitz ◽  
Massimiliano Marangio ◽  
Anja Pruchnewski ◽  
Margit Voigt
Keyword(s):  
Choice Number ◽  
Sum Choice Number

Locally-iterative Distributed (Δ + 1)-coloring and Applications

2022 ◽  
Vol 69 (1) ◽  
pp. 1-26
Author(s):  
Leonid Barenboim ◽  
Michael Elkin ◽  
Uri Goldenberg
Keyword(s):  
Message Passing ◽  
Edge Coloring ◽  
Iterative Algorithms ◽  
Independent Set ◽  
Local Model ◽  
Vertex Coloring ◽  
Maximal Independent Set ◽  
List Type ◽  
Inherent Limitation ◽  
Running Time

We consider graph coloring and related problems in the distributed message-passing model. Locally-iterative algorithms are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1-hop-neighborhood . In STOC’93 Szegedy and Vishwanathan showed that any locally-iterative Δ + 1-coloring algorithm requires Ω (Δ log Δ + log * n ) rounds, unless there exists “a very special type of coloring that can be very efficiently reduced” [ 44 ]. No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms and to explore other approaches to the coloring problem [ 2 , 3 , 19 , 32 ]. The latter gave rise to faster algorithms, but their heavy machinery that is of non-locally-iterative nature made them far less suitable to various settings. In this article, we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative Δ + 1-coloring algorithm with running time O (Δ + log * n ), i.e., below Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing, and bandwidth-restricted settings. This includes the following results: We obtain self-stabilizing distributed algorithms for Δ + 1-vertex-coloring, (2Δ - 1)-edge-coloring, maximal independent set, and maximal matching with O (Δ + log * n ) time. This significantly improves previously known results that have O(n) or larger running times [ 23 ]. We devise a (2Δ - 1)-edge-coloring algorithm in the CONGEST model with O (Δ + log * n ) time and O (Δ)-edge-coloring in the Bit-Round model with O (Δ + log n ) time. The factors of log * n and log n are unavoidable in the CONGEST and Bit-Round models, respectively. Previously known algorithms had superlinear dependency on Δ for (2Δ - 1)-edge-coloring in these models. We obtain an arbdefective coloring algorithm with running time O (√ Δ + log * n ). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it to compute a proper (1 + ε)Δ-coloring within O (√ Δ + log * n ) time and Δ + 1-coloring within O (√ Δ log Δ log * Δ + log * n ) time. This improves the recent state-of-the-art bounds of Barenboim from PODC’15 [ 2 ] and Fraigniaud et al. from FOCS’16 [ 19 ] by polylogarithmic factors. Our algorithms are applicable to the SET-LOCAL model [ 25 ] (also known as the weak LOCAL model). In this model a relatively strong lower bound of Ω (Δ 1/3 ) is known for Δ + 1-coloring. However, most of the coloring algorithms do not work in this model. (In Reference [ 25 ] only Linial’s O (Δ 2 )-time algorithm and Kuhn-Wattenhofer O (Δ log Δ)-time algorithms are shown to work in it.) We obtain the first linear-in-Δ Δ + 1-coloring algorithms that work also in this model.

scholarly journals Grundy Number of Some Chordal Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 64
Author(s):  
R. Nagarathinam ◽  
N. Parvathi ◽  
. .
Keyword(s):  
Line Graph ◽  
Cartesian Product ◽  
Chordal Graphs ◽  
Complete Graphs ◽  
Proper Coloring ◽  
Coloring Problem ◽  
Grundy Number

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c : V → {1, 2, . . .} such that c(u) 12≠"> c(v) for every edge uv ∈ E. For proper coloring, colors assigned must be minimum, but for Grundy coloring which should be maximum. In this instance, Grundy numbers of chordal graphs like Cartesian product of two path graphs, join of the path and complete graphs and the line graph of tadpole have been executed 

Sign in / Sign up

Export Citation Format

Share Document