scholarly journals On-line coloring of $I_s$-free graphs

2005 ◽  
Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽  
Author(s):  
Iwona Cieslik ◽  
Marcin Kozik ◽  
Piotr Micek
Keyword(s):  
Greedy Algorithm ◽  
Planar Graphs ◽  
Greedy Algorithms ◽  
Independent Set ◽  
Vertex Coloring ◽  
Tight Bound ◽  
Free Graph ◽  
Coloring Problem ◽  
International Audience ◽  
On Line

International audience An on-line vertex coloring algorithm receives vertices of a graph in some externally determined order. Each new vertex is presented together with a set of the edges connecting it to the previously presented vertices. As a vertex is presented, the algorithm assigns it a color which cannot be changed afterwards. The on-line coloring problem was addressed for many different classes of graphs defined in terms of forbidden structures. We analyze the class of $I_s$-free graphs, i.e., graphs in which the maximal size of an independent set is at most $s-1$. An old Szemerédi's result implies that for each on-line algorithm A there exists an on-line presentation of an $I_s$-free graph $G$ forcing A to use at least $\frac{s}{2}χ ^{(G)}$ colors. We prove that any greedy algorithm uses at most $\frac{s}{2}χ^{(G)}$ colors for any on-line presentation of any $I_s$-free graph $G$. Since the class of co-planar graphs is a subclass of $I_5$-free graphs all greedy algorithms use at most $\frac{5}{2}χ (G)$ colors for co-planar $G$'s. We prove that, even in a smaller class, this is an almost tight bound.

scholarly journals Building a Maximal Independent Set for the Vertex-coloring Problem on Planar Graphs

2020 ◽  
Vol 354 ◽  
pp. 75-89
Author(s):  
Cristina López-Ramírez ◽  
Jorge Eduardo Gutiérrez Gómez ◽  
Guillermo De Ita Luna
Keyword(s):  
Planar Graphs ◽  
Independent Set ◽  
Vertex Coloring ◽  
Maximal Independent Set ◽  
Coloring Problem ◽  
Vertex Coloring Problem

scholarly journals An improved bound on the largest induced forests for triangle-free planar graphs

2010 ◽  
Vol Vol. 12 no. 1 ◽  
Author(s):  
Lukasz Kowalik ◽  
Borut Luzar ◽  
Riste Skrekovski
Keyword(s):  
Planar Graphs ◽  
Free Graph ◽  
International Audience

International audience We proved that every planar triangle-free graph of order n has a subset of vertices that induces a forest of size at least (71n + 72)/128. This improves the earlier work of Salavatipour (2006). We also pose some questions regarding planar graphs of higher girth.

scholarly journals Split-critical and uniquely split-colorable graphs

2010 ◽  
Vol Vol. 12 no. 5 (Graph and Algorithms) ◽  
Author(s):  
Tınaz Ekim ◽  
Bernard Ries ◽  
Dominique De Werra
Keyword(s):  
Chromatic Number ◽  
Point Of View ◽  
Vertex Coloring ◽  
Research Directions ◽  
Coloring Problem ◽  
Minimum Number ◽  
Vertex Coloring Problem ◽  
International Audience ◽  
Split Graphs

Graphs and Algorithms International audience The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some properties of split-critical and uniquely split-colorable graphs. We describe constructions of such graphs with some additional properties. We also study the effect of the addition and the removal of some edge sets on the value of the split-chromatic number. All these results are compared with their cochromatic counterparts. We conclude with several research directions on the topic.

scholarly journals Matroids And Greedy Algorithms. A Deeper Justification of Using Greedy Approach To Find A Maximal set of a Matroid

2017 ◽  
Vol 16 (2) ◽  
pp. 48
Author(s):  
Syed Aqib Haider
Keyword(s):  
Greedy Algorithm ◽  
Local Level ◽  
Greedy Algorithms ◽  
Optimal Solution ◽  
Independent Set ◽  
Computation Time ◽  
Optimal Choice ◽  
Greedy Approach ◽  
The Greedy Algorithm ◽  
Selection Of

<p>Greedy algorithms are used in solving a diverse set of problems in small computation time. However, for solving problems using greedy approach, it must be proved that the greedy strategy applies. The greedy approach relies on selection of optimal choice at a local level reducing the problem to a single sub problem, which actually leads to a globally optimal solution. Finding a maximal set from the independent set of a matroid M(S, I) also uses greedy approach and justification is also provided in standard literature (e.g. Introduction to Algorithms by Cormen et .al.). However, the justification does not clearly explain the equivalence of using greedy algorithm and contraction of M by the selected element. This paper thus attempts to give a lucid explanation of the fact that the greedy algorithm is equivalent to reducing the Matroid into its contraction by selected element. This approach also provides motivation for research on the selection of the test used in algorithm which might lead to smaller computation time of the algorithm.</p>

scholarly journals A new efficient RLF-like algorithm for the vertex coloring problem

2016 ◽  
Vol 26 (4) ◽  
pp. 441-456 ◽  
Author(s):  
Mourchid Adegbindin ◽  
Alain Hertz ◽  
Martine Bellaïche
Keyword(s):  
Greedy Algorithm ◽  
Chromatic Number ◽  
Upper Bound ◽  
Computational Experiments ◽  
Vertex Coloring ◽  
Greedy Heuristics ◽  
Selection Rules ◽  
Coloring Problem ◽  
Color Class ◽  
Vertex Coloring Problem

The Recursive Largest First (RLF) algorithm is one of the most popular greedy heuristics for the vertex coloring problem. It sequentially builds color classes on the basis of greedy choices. In particular, the first vertex placed in a color class C is one with a maximum number of uncolored neighbors, and the next vertices placed in C are chosen so that they have as many uncolored neighbors which cannot be placed in C. These greedy choices can have a significant impact on the performance of the algorithm, which explains why we propose alternative selection rules. Computational experiments on 63 difficult DIMACS instances show that the resulting new RLF-like algorithm, when compared with the standard RLF, allows to obtain a reduction of more than 50% of the gap between the number of colors used and the best known upper bound on the chromatic number. The new greedy algorithm even competes with basic metaheuristics for the vertex coloring problem.

scholarly journals Nonrepetitive colorings of graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Noga Alon ◽  
Jaroslaw Grytczuk
Keyword(s):  
20Th Century ◽  
Open Problem ◽  
Planar Graphs ◽  
Periodic Sequence ◽  
Simple Path ◽  
Vertex Coloring ◽  
Graph Colorings ◽  
Interesting Open Problem ◽  
Minimum Number ◽  
International Audience

International audience A vertex coloring of a graph $G$ is $k \textit{-nonrepetitive}$ if one cannot find a periodic sequence with $k$ blocks on any simple path of $G$. The minimum number of colors needed for such coloring is denoted by $\pi _k(G)$ . This idea combines graph colorings with Thue sequences introduced at the beginning of 20th century. In particular Thue proved that if $G$ is a simple path of any length greater than $4$ then $\pi _2(G)=3$ and $\pi_3(G)=2$. We investigate $\pi_k(G)$ for other classes of graphs. Particularly interesting open problem is to decide if there is, possibly huge, $k$ such that $\pi_k(G)$ is bounded for planar graphs.

scholarly journals Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections

2009 ◽  
Vol 19 (1) ◽  
pp. 61-85 ◽  
Author(s):  
DAVID GAMARNIK ◽  
DAVID A. GOLDBERG
Keyword(s):  
Greedy Algorithm ◽  
Greedy Algorithms ◽  
Independent Set ◽  
Independent Sets ◽  
Simple Expression ◽  
Regular Graphs ◽  
Bounded Degree ◽  
Constant Degree ◽  
Image Position ◽  
The Greedy Algorithm

We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant-degree regular graphs. We show that forr-regular graphs withnnodes and girth at leastg, the algorithm finds an independent set of expected cardinalitywheref(r) is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degreerand girthgare large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm inarbitrarybounded-degree graphs is concentrated around the mean. Finally, we analyse the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.

scholarly journals Linear choosability of graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Louis Esperet ◽  
Mickael Montassier ◽  
André Raspaud
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Oriented Graph ◽  
Vertex Coloring ◽  
Proper Coloring ◽  
Maximum Average Degree ◽  
Complete Problem ◽  
International Audience ◽  
Np Complete ◽  
Proper Vertex

International audience A proper vertex coloring of a non oriented graph $G=(V,E)$ is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph $G$ is $L$-list colorable if for a given list assignment $L=\{L(v): v∈V\}$, there exists a proper coloring $c$ of $G$ such that $c(v)∈L(v)$ for all $v∈V$. If $G$ is $L$-list colorable for every list assignment with $|L(v)|≥k$ for all $v∈V$, then $G$ is said $k$-choosable. A graph is said to be lineary $k$-choosable if the coloring obtained is linear. In this paper, we investigate the linear choosability of graphs for some families of graphs: graphs with small maximum degree, with given maximum average degree, planar graphs... Moreover, we prove that determining whether a bipartite subcubic planar graph is lineary 3-colorable is an NP-complete problem.

scholarly journals 8-star-choosability of a graph with maximum average degree less than 3

2011 ◽  
Vol Vol. 13 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Min Chen ◽  
André Raspaud ◽  
Weifan Wang
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Average Degree ◽  
Vertex Coloring ◽  
Maximum Average Degree ◽  
International Audience ◽  
Star Coloring ◽  
List Chromatic Number ◽  
Proper Vertex

Graphs and Algorithms International audience A proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) epsilon L(v). If G is L-star-colorable for any list assignment L with vertical bar L(v)vertical bar \textgreater= k for all v epsilon V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by X-s(l)(G), is the smallest integer k such that G is k-star-choosable. In this article, we prove that every graph G with maximum average degree less than 3 is 8-star-choosable. This extends a result that planar graphs of girth at least 6 are 8-star-choosable [A. Kundgen, C. Timmons, Star coloring planar graphs from small lists, J. Graph Theory, 63(4): 324-337, 2010].

A DNA computer model for solving vertex coloring problem

2006 ◽  
Vol 51 (20) ◽  
pp. 2541-2549 ◽  
Author(s):  
Jin Xu ◽  
Xiaoli Qiang ◽  
Fang Gang ◽  
Kang Zhou
Keyword(s):  
Computer Model ◽  
Vertex Coloring ◽  
Coloring Problem ◽  
Vertex Coloring Problem ◽  
Dna Computer
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