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.

Self-Stabilizing Graph Coloring Algorithms

2013 ◽  
pp. 96-110
Author(s):  
Shing-Tsaan Huang ◽  
Chi-Hung Tzeng ◽  
Jehn-Ruey Jiang
Keyword(s):  
Distributed Computing ◽  
Graph Coloring ◽  
Finite Time ◽  
Planar Graphs ◽  
Edge Coloring ◽  
Vertex Coloring ◽  
Transient Faults ◽  
Initial State ◽  
Self Stabilization ◽  
Legitimate State

The concept of self-stabilization in distributed systems was introduced by Dijkstra in 1974. A system is said to be self-stabilizing if (1) it can converge in finite time to a legitimate state from any initial state, and (2) when it is in a legitimate state, it remains so henceforth. That is, a self-stabilizing system guarantees to converge to a legitimate state in finite time no matter what initial state it may start with; or, it can recover from transient faults automatically without any outside intervention. This chapter first introduces the self-stabilization concept in distributed computing. Next, it discusses the coloring problem on graphs and its applications in distributed computing. Then, it introduces three self-stabilizing algorithms. The first two are for vertex coloring and edge coloring on planar graphs, respectively. The last one is for edge coloring on bipartite graphs.

(1,0)-Relaxed strong edge list coloring of planar graphs with girth 6

2019 ◽  
Vol 11 (06) ◽  
pp. 1950064
Author(s):  
Kai Lin ◽  
Min Chen ◽  
Dong Chen
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring ◽  
List Coloring ◽  
Chromatic Index ◽  
Strong Chromatic Index ◽  
Minimum Number ◽  
Number Formula ◽  
Edge List

Let [Formula: see text] be a graph. An [Formula: see text]-relaxed strong edge [Formula: see text]-coloring is a mapping [Formula: see text] such that for any edge [Formula: see text], there are at most [Formula: see text] edges adjacent to [Formula: see text] and [Formula: see text] edges which are distance two apart from [Formula: see text] assigned the same color as [Formula: see text]. The [Formula: see text]-relaxed strong chromatic index, denoted by [Formula: see text], is the minimum number [Formula: see text] of an [Formula: see text]-relaxed strong [Formula: see text]-edge-coloring admitted by [Formula: see text]. [Formula: see text] is called [Formula: see text]-relaxed strong edge [Formula: see text]-colorable if for a given list assignment [Formula: see text], there exists an [Formula: see text]-relaxed strong edge coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text]. If [Formula: see text] is [Formula: see text]-relaxed strong edge [Formula: see text]-colorable for any list assignment with [Formula: see text] for all [Formula: see text], then [Formula: see text] is said to be [Formula: see text]-relaxed strong edge [Formula: see text]-choosable. The [Formula: see text]-relaxed strong list chromatic index, denoted by [Formula: see text], is defined to be the smallest integer [Formula: see text] such that [Formula: see text] is [Formula: see text]-relaxed strong edge [Formula: see text]-choosable. In this paper, we prove that every planar graph [Formula: see text] with girth 6 satisfies that [Formula: see text]. This strengthens a result which says that every planar graph [Formula: see text] with girth 7 and [Formula: see text] satisfies that [Formula: see text].

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 $n^2 + \lceil 5n/3 \rceil$.

A STUDY ON FUZZY EQUITABLE EDGE COLORING OF WHEEL RELATED GRAPHS

2020 ◽  
Vol 9 (11) ◽  
pp. 9311-9317
Author(s):  
K. Sivaraman ◽  
R.V. Prasad
Keyword(s):  
Edge Coloring ◽  
Graph Labeling ◽  
Equitable Edge Coloring

Equitable edge coloring is a kind of graph labeling with the following restrictions. No two adjacent edges receive same label (color). and number of edges in any two color classes differ by at most one. In this work we are going to present the Fuzzy equitable edge coloring of some wheel related graphs.

scholarly journals Canine Malignant Mammary Tumours II. Adenocarcinomas, Solid Carcinomas and Spindle Cell Carcinomas

1972 ◽  
Vol 9 (6) ◽  
pp. 447-470 ◽  
Author(s):  
W. Misdorp ◽  
E. Cotchin ◽  
J. F. Hampe ◽  
Anne G. Jabara ◽  
J. von Sandersleben
Keyword(s):  
Spindle Cell ◽  
Clinical Signs ◽  
Subjective Assessment ◽  
Simple Type ◽  
Well Differentiated ◽  
The Times ◽  
Lymphatic Permeation ◽  
Preliminary Classification ◽  
Mammary Adenocarcinomas

A preliminary classification of 130 canine mammary adenocarcinomas, 76 solid carcinomas, and nine spindle cell carcinomas, together with several subtypes, was constructed from pooled, selected (metastasized) material. Each tumour in this series was classified by subjective assessment of its quantitatively predominant histological picture. Many adenocarcinomas and solid carcinomas of simple type were infiltrative, and lymphatic permeation was often found. The complex types of adenocarcinomas and of solid carcinomas were expansive, and lymphatic permeation was rare. Some metastasized adenocarcinomas were well differentiated. The clinical signs, distribution of metastases and some preliminary data on the times of survival of dogs with various types of carcinomas are discussed.

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.

Complexity classification of the edge coloring problem for a family of graph classes

2017 ◽  
Vol 27 (2) ◽  
Author(s):  
Dmitriy S. Malyshev
Keyword(s):  
Edge Coloring ◽  
Complete Classification ◽  
Chromatic Index ◽  
Forbidden Subgraphs ◽  
Coloring Problem ◽  
Graph Classes ◽  
Complexity Classification ◽  
Index Problem

AbstractA class of graphs is called monotone if it is closed under deletion of vertices and edges. Any such class may be defined in terms of forbidden subgraphs. The chromatic index of a graph is the smallest number of colors required for its edge-coloring such that any two adjacent edges have different colors. We obtain a complete classification of the complexity of the chromatic index problem for all monotone classes defined in terms of forbidden subgraphs having at most 6 edges or at most 7 vertices.

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