scholarly journals The Chromatic Number of Two Families of Generalized Kneser Graphs Related to Finite Generalized Quadrangles and Finite Projective 3-Spaces

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Klaus Metsch
Keyword(s):  
Projective Space ◽  
Chromatic Number ◽  
General Position ◽  
Independence Number ◽  
Independent Set ◽  
Generalized Quadrangle ◽  
Finite Projective Space ◽  
Maximal Independent Set ◽  
Generalized Quadrangles ◽  
Kneser Graphs

Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective space $\mathrm{PG}(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\geqslant 43$ we determine the largest independent set of $\Gamma$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\geqslant 47$, this information is then used to show that $\Gamma$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove similar results for the graph that is built in the same way on the chambers of the generalized quadrangle.

VERTEX VULNERABILITY PARAMETER OF GEAR GRAPHS

2011 ◽  
Vol 22 (05) ◽  
pp. 1187-1195 ◽  
Author(s):  
AYSUN AYTAC ◽  
TUFAN TURACI
Keyword(s):  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Maximal Independent Set ◽  
Minimum Cardinality ◽  
Independent Domination

For a vertex v of a graph G = (V,E), the independent domination number (also called the lower independence number) iv(G) of G relative to v is the minimum cardinality of a maximal independent set in G that contains v. The average lower independence number of G is [Formula: see text]. In this paper, this parameter is defined and examined, also the average lower independence number of gear graphs is considered. Then, an algorithm for the average lower independence number of any graph is offered.

scholarly journals On the Cartesian Product of Non Well-Covered Graphs

10.37236/2299 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Bert L Hartnell ◽  
Douglas F Rall
Keyword(s):  
Independence Number ◽  
Cartesian Product ◽  
Independent Set ◽  
Maximal Independent Set

A graph is well-covered if every maximal independent set has the same cardinality, namely the vertex independence number.  We answer a question of Topp and Volkmann and prove that if the Cartesian product of two graphs is well-covered, then at least one of them must be well-covered.

scholarly journals Operations on Well-Covered Graphs and the Roller-Coaster Conjecture

10.37236/1798 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Philip Matchett
Keyword(s):  
Positive Integer ◽  
Simple Proof ◽  
Linear Order ◽  
Independence Number ◽  
Independent Set ◽  
Maximal Independent Set ◽  
Roller Coaster ◽  
Graph Theoretic ◽  
Linear Orderings ◽  
Independence Polynomial

A graph $G$ is well-covered if every maximal independent set has the same cardinality. Let $s_k$ denote the number of independent sets of cardinality $k$, and define the independence polynomial of $G$ to be $S(G,z) = \sum s_kz^k$. This paper develops a new graph theoretic operation called power magnification that preserves well-coveredness and has the effect of multiplying an independence polynomial by $z^c$ where $c$ is a positive integer. We will apply power magnification to the recent Roller-Coaster Conjecture of Michael and Traves, proving in our main theorem that for sufficiently large independence number $\alpha$, it is possible to find well-covered graphs with the last $(.17)\alpha$ terms of the independence sequence in any given linear order. Also, we will give a simple proof of a result due to Alavi, Malde, Schwenk, and Erdős on possible linear orderings of the independence sequence of not-necessarily well-covered graphs, and we will prove the Roller-Coaster Conjecture in full for independence number $\alpha \leq 11$. Finally, we will develop two new graph operations that preserve well-coveredness and have interesting effects on the independence polynomial.

scholarly journals Competition-Independence Game and Domination Game

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 359 ◽  
Author(s):  
Chalermpong Worawannotai ◽  
Watcharintorn Ruksasakchai
Keyword(s):  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Maximal Independent Set ◽  
Optimal Size ◽  
Independence Numbers ◽  
Game Domination Number ◽  
Domination Game ◽  
Domination Numbers

The domination game is played on a graph by two players, Dominator and Staller, who alternately choose a vertex of G. Dominator aims to finish the game in as few turns as possible while Staller aims to finish the game in as many turns as possible. The game ends when all vertices are dominated. The game domination number, denoted by γ g ( G ) (respectively γ g ′ ( G ) ), is the total number of turns when both players play optimally and when Dominator (respectively Staller) starts the game. In this paper, we study a version of this game where the set of chosen vertices is always independent. This version turns out to be another game known as the competition-independence game. The competition-independence game is played on a graph by two players, Diminisher and Sweller. They take turns in constructing maximal independent set M, where Diminisher tries to minimize | M | and Sweller tries to maximize | M | . Note that, actually, it is the domination game in which the set of played vertices is independent. The competition-independence number, denoted by I d ( G ) (respectively I s ( G ) ) is the optimal size of the final independent set in the competition-independence game if Diminisher (respectively Sweller) starts the game. In this paper, we check whether some well-known results in the domination game hold for the competition-independence game. We compare the competition-independence numbers to the game domination numbers. Moreover, we provide a family of graphs such that many parameters are equal. Finally, we present a realization result on the competition-independence numbers.

Roman domination and 2-independence in trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Nacéra Meddah ◽  
Mustapha Chellali
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Sum Formula ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

A High-Quality and Fast Maximal Independent Set Implementation for GPUs

2019 ◽  
Vol 5 (2) ◽  
pp. 1-27
Author(s):  
Martin Burtscher ◽  
Sindhu Devale ◽  
Sahar Azimi ◽  
Jayadharini Jaiganesh ◽  
Evan Powers
Keyword(s):  
Independent Set ◽  
Maximal Independent Set ◽  
High Quality

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

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