Turán-Ramsey Theorems and Kp-Independence Numbers

1994 ◽  
Vol 3 (3) ◽  
pp. 297-325 ◽  
Author(s):  
P. Erdős ◽  
A. Hajnal ◽  
M. Simonovits ◽  
V. T. Sós ◽  
E. Szemerédi
Keyword(s):  
Simple Structure ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Size ◽  
Maximum Order ◽  
Induced Subgraph ◽  
Asymptotic Value ◽  
Image Position ◽  
Independence Numbers ◽  
Edge Colouring

Let the Kp-independence number αp (G) of a graph G be the maximum order of an induced subgraph in G that contains no Kp. (So K2-independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L1,…,Lr, we define the corresponding Turán-Ramsey function RTp(n, L1,…,Lr, m) to be the maximum number of edges in a graph Gn of order n such that αp(Gn) ≤ m and there is an edge-colouring of G with r colours such that the jth colour class contains no copy of Lj, for j = 1,…, r. In this continuation of [11] and [12], we will investigate the problem where, instead of α(Gn) = o(n), we assume (for some fixed p > 2) the stronger condition that αp(Gn) = o(n). The first part of the paper contains multicoloured Turán-Ramsey theorems for graphs Gn of order n with small Kp-independence number αp(Gn). Some structure theorems are given for the case αp(Gn) = o(n), showing that there are graphs with fairly simple structure that are within o(n2) of the extremal size; the structure is described in terms of the edge densities between certain sets of vertices.The second part of the paper is devoted to the case r = 1, i.e., to the problem of determining the asymptotic value offor p < q. Several results are proved, and some other problems and conjectures are stated.

Inductive graph invariants and approximation algorithms

2021 ◽  
Author(s):  
C. R. Subramanian
Keyword(s):  
Optimization Problems ◽  
Independence Number ◽  
Optimal Solution ◽  
Maximum Size ◽  
Chordal Graphs ◽  
Minimum Size ◽  
Optimal Size ◽  
Induced Subgraph ◽  
Special Cases ◽  
Optimal Formula

We introduce and study an inductively defined analogue [Formula: see text] of any increasing graph invariant [Formula: see text]. An invariant [Formula: see text] is increasing if [Formula: see text] whenever [Formula: see text] is an induced subgraph of [Formula: see text]. This inductive analogue simultaneously generalizes and unifies known notions like degeneracy, inductive independence number, etc., into a single generic notion. For any given increasing [Formula: see text], this gets us several new invariants and many of which are also increasing. It is also shown that [Formula: see text] is the minimum (over all orderings) of a value associated with each ordering. We also explore the possibility of computing [Formula: see text] (and a corresponding optimal vertex ordering) and identify some pairs [Formula: see text] for which [Formula: see text] can be computed efficiently for members of [Formula: see text]. In particular, it includes graphs of bounded [Formula: see text] values. Some specific examples (like the class of chordal graphs) have already been studied extensively. We further extend this new notion by (i) allowing vertex weighted graphs, (ii) allowing [Formula: see text] to take values from a totally ordered universe with a minimum and (iii) allowing the consideration of [Formula: see text]-neighborhoods for arbitrary but fixed [Formula: see text]. Such a generalization is employed in designing efficient approximations of some graph optimization problems. Precisely, we obtain efficient algorithms (by generalizing the known algorithm of Ye and Borodin [Y. Ye and A. Borodin, Elimination graphs, ACM Trans. Algorithms 8(2) (2012) 1–23] for special cases) for approximating optimal weighted induced [Formula: see text]-subgraphs and optimal [Formula: see text]-colorings (for hereditary [Formula: see text]’s) within multiplicative factors of (essentially) [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] denotes the inductive analogue (as defined in this work) of optimal size of an unweighted induced [Formula: see text]-subgraph of the input and [Formula: see text] is the minimum size of a forbidden induced subgraph of [Formula: see text]. Our results generalize the previous result on efficiently approximating maximum independent sets and minimum colorings on graphs of bounded inductive independence number to optimal [Formula: see text]-subgraphs and [Formula: see text]-colorings for arbitrary hereditary classes [Formula: see text]. As a corollary, it is also shown that any maximal [Formula: see text]-subgraph approximates an optimal solution within a factor of [Formula: see text] for unweighted graphs, where [Formula: see text] is maximum size of any induced [Formula: see text]-subgraph in any local neighborhood [Formula: see text].

scholarly journals On the k -Component Independence Number of a Tree

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Shuting Cheng ◽  
Baoyindureng Wu
Keyword(s):  
Linear Time ◽  
Independence Number ◽  
Independent Set ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Maximum Order ◽  
Component Order

Let G be a graph and k ≥ 1 be an integer. A subset S of vertices in a graph G is called a k -component independent set of G if each component of G S has order at most k . The k -component independence number, denoted by α c k G , is the maximum order of a vertex subset that induces a subgraph with maximum component order at most k . We prove that if a tree T is of order n , then α k T ≥ k / k + 1 n . The bound is sharp. In addition, we give a linear-time algorithm for finding a maximum k -component independent set of a tree.

On the Independence Number of Steiner Systems

2013 ◽  
Vol 22 (2) ◽  
pp. 241-252 ◽  
Author(s):  
ALEX EUSTIS ◽  
JACQUES VERSTRAËTE
Keyword(s):  
Independence Number ◽  
Maximum Size ◽  
Uniform Hypergraph ◽  
Steiner Systems ◽  
L System ◽  
Formula Group ◽  
Image Position

Apartial Steiner (n,r,l)-systemis anr-uniform hypergraph onnvertices in which every set oflvertices is contained in at most one edge. A partial Steiner (n,r,l)-system iscompleteif every set oflvertices is contained in exactly one edge. In a hypergraph, the independence number α() denotes the maximum size of a set of vertices incontaining no edge. In this article we prove the following. Given integersr,lsuch thatr≥ 2l− 1 ≥ 3, we prove that there exists a partial Steiner (n,r,l)-systemsuch that$$\alpha(\HH) \lesssim \biggl(\frac{l-1}{r-1}(r)_l\biggr)^{\frac{1}{r-1}}n^{\frac{r-l}{r-1}} (\log n)^{\frac{1}{r-1}} \quad \mbox{ as }n \rightarrow \infty.$$This improves earlier results of Phelps and Rödl, and Rödl and Ŝinajová. We conjecture that it is best possible as it matches the independence number of a randomr-uniform hypergraph of the same density. Ifl= 2 orl= 3, then for infinitely manyrthe partial Steiner systems constructed are complete for infinitely manyn.

Exponential Independence Number of Some Graphs

2018 ◽  
Vol 29 (07) ◽  
pp. 1151-1164
Author(s):  
Canan Çiftçi ◽  
Aysun Aytaç
Keyword(s):  
Shortest Path ◽  
Independence Number ◽  
General Theorem ◽  
Independent Set ◽  
Maximum Order ◽  
Tree Graphs ◽  
Number Formula ◽  
Thorn Graph

Let [Formula: see text] be a graph and [Formula: see text]. We define by [Formula: see text] the subgraph of [Formula: see text] induced by [Formula: see text]. For each vertex [Formula: see text] and for each vertex [Formula: see text], [Formula: see text] is the length of the shortest path in [Formula: see text] between [Formula: see text] and [Formula: see text] if such a path exists, and [Formula: see text] otherwise. For a vertex [Formula: see text], let [Formula: see text] where [Formula: see text]. Jäger and Rautenbach [27] define a set [Formula: see text] of vertices to be exponential independent if [Formula: see text] for every vertex [Formula: see text] in [Formula: see text]. The exponential independence number [Formula: see text] of [Formula: see text] is the maximum order of an exponential independent set. In this paper, we give a general theorem and we examine exponential independence number of some tree graphs and thorn graph of some graphs.

scholarly journals On Some Conjectures Concerning Critical Independent Sets of a Graph

10.37236/5580 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Taylor Short
Keyword(s):  
Lower Bound ◽  
Independence Number ◽  
Independent Set ◽  
Maximum Size ◽  
Independent Sets ◽  
Simple Graph ◽  
Vertex Set ◽  
The Difference

Let $G$ be a simple graph with vertex set $V(G)$. A set $S\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. For $X\subseteq V(G)$, the difference of $X$ is $d(X) = |X|-|N(X)|$ and an independent set $A$ is critical if $d(A) = \max \{d(X): X\subseteq V(G) \text{ is an independent set}\}$ (possibly $A=\emptyset$). Let $\text{nucleus}(G)$ and $\text{diadem}(G)$ be the intersection and union, respectively, of all maximum size critical independent sets in $G$. In this paper, we will give two new characterizations of Konig-Egervary graphs involving $\text{nucleus}(G)$ and $\text{diadem}(G)$. We also prove a related lower bound for the independence number of a graph. This work answers several conjectures posed by Jarden, Levit, and Mandrescu.

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.

scholarly journals ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

2021 ◽  
pp. 1-40
Author(s):  
NICK GILL ◽  
BIANCA LODÀ ◽  
PABLO SPIGA
Keyword(s):  
Permutation Group ◽  
Model Theory ◽  
Independent Set ◽  
Maximum Size ◽  
Proper Subset ◽  
Permutation Groups ◽  
Finite Permutation Group ◽  
Pointwise Stabilizer ◽  
Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

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.

scholarly journals Planar Graphs have Independence Ratio at least 3/13

10.37236/5309 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Independence Number ◽  
Independent Set

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$.  In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

A Note on Coverings of En by Convex Sets

1967 ◽  
Vol 10 (5) ◽  
pp. 669-673 ◽  
Author(s):  
J.H.H. Chalk
Keyword(s):  
Convex Sets ◽  
Independent Set ◽  
Linearly Independent ◽  
Image Position ◽  
Take All

Let (x1, x2, …, xn) denote the coordinates of a point of Euclidean n-space En. Let be a set of n+1 points of En with the property thatform a linearly independent set and define a lattice Λ of pointsby allowing u1, …, un to take all integer values.

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