Independent Sets in the Union of Two Hamiltonian Cycles

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c<c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t <c$, $f(n,cn)$ is exponentially large in $n ~(n \to \infty)$. We prove that $0.26 < c_t < 0.36$, but the exact value of $c_t$ is not determined. For the lower bound we prove a technical lemma, which for graphs that are the union of two Hamiltonian cycles establishes a relation between the independence number and the number of $K_4$ subgraphs. A corollary of this lemma is that if a graph $G$ on $n>12$ vertices is the union of two Hamiltonian cycles and $\alpha(G)=n/4$, then $V(G)$ can be covered by vertex-disjoint $K_4$ subgraphs.

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New bounds on the independence number of connected graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500696 ◽

2018 ◽

Vol 10 (05) ◽

pp. 1850069

Author(s):

Nader Jafari Rad ◽

Elahe Sharifi

Keyword(s):

Lower Bound ◽

Connected Graph ◽

Maximum Degree ◽

Applied Mathematics ◽

Independence Number ◽

Independent Set ◽

Maximum Cardinality ◽

Connected Graphs ◽

Cut Vertex

The independence number of a graph [Formula: see text], denoted by [Formula: see text], is the maximum cardinality of an independent set of vertices in [Formula: see text]. [Henning and Löwenstein An improved lower bound on the independence number of a graph, Discrete Applied Mathematics 179 (2014) 120–128.] proved that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] does not belong to a specific family of graphs, then [Formula: see text]. In this paper, we strengthen the above bound for connected graphs with maximum degree at least three that have a non-cut-vertex of maximum degree. We show that if a connected graph [Formula: see text] of order [Formula: see text] and size [Formula: see text] has a non-cut-vertex of maximum degree then [Formula: see text], where [Formula: see text] is the maximum degree of the vertices of [Formula: see text]. We also characterize all connected graphs [Formula: see text] of order [Formula: see text] and size [Formula: see text] that have a non-cut-vertex of maximum degree and [Formula: see text].

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A New Lower Bound on the Independence Number of a Graph and Applications

The Electronic Journal of Combinatorics ◽

10.37236/3601 ◽

2014 ◽

Vol 21 (1) ◽

Cited By ~ 7

Author(s):

Michael A. Henning ◽

Christian Löwenstein ◽

Justin Southey ◽

Anders Yeo

Keyword(s):

Lower Bound ◽

Maximum Degree ◽

Independence Number ◽

Maximum Clique ◽

Independent Set ◽

Uniform Hypergraph ◽

Maximum Cardinality ◽

Uniform Hypergraphs ◽

Transversal Number ◽

Graph Parameters

The independence number of a graph $G$, denoted $\alpha(G)$, is the maximum cardinality of an independent set of vertices in $G$. The independence number is one of the most fundamental and well-studied graph parameters. In this paper, we strengthen a result of Fajtlowicz [Combinatorica 4 (1984), 35-38] on the independence of a graph given its maximum degree and maximum clique size. As a consequence of our result we give bounds on the independence number and transversal number of $6$-uniform hypergraphs with maximum degree three. This gives support for a conjecture due to Tuza and Vestergaard [Discussiones Math. Graph Theory 22 (2002), 199-210] that if $H$ is a $3$-regular $6$-uniform hypergraph of order $n$, then $\tau(H) \le n/4$.

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TRANSFERENCE FOR THE ERDŐS–KO–RADO THEOREM

Forum of Mathematics Sigma ◽

10.1017/fms.2015.21 ◽

2015 ◽

Vol 3 ◽

Cited By ~ 6

Author(s):

JÓZSEF BALOGH ◽

BÉLA BOLLOBÁS ◽

BHARGAV P. NARAYANAN

Keyword(s):

Random Graph ◽

Independence Number ◽

Independent Set ◽

Independent Interest ◽

Natural Numbers ◽

Kneser Graph ◽

Intersecting Family ◽

The Family

For natural numbers $n,r\in \mathbb{N}$ with $n\geqslant r$, the Kneser graph $K(n,r)$ is the graph on the family of $r$-element subsets of $\{1,\ldots ,n\}$ in which two sets are adjacent if and only if they are disjoint. Delete the edges of $K(n,r)$ with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as $r/n$ is bounded away from $1/2$, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős–Ko–Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.

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New Approach to the $k$-Independence Number of a Graph

The Electronic Journal of Combinatorics ◽

10.37236/2646 ◽

2013 ◽

Vol 20 (1) ◽

Cited By ~ 13

Author(s):

Yair Caro ◽

Adriana Hansberg

Keyword(s):

Graph Theory ◽

Lower Bound ◽

Lower Bounds ◽

Maximum Degree ◽

Independence Number ◽

Independent Set ◽

Average Degree ◽

Maximum Cardinality ◽

New Approach

Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$-independent set $S \subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\alpha_k(G)$ we denote the maximum cardinality of a $k$-independent set of $G$. We prove that, for a graph $G$ on $n$ vertices and average degree $d$, $\alpha_k(G) \ge \frac{k+1}{\lceil d \rceil + k + 1} n$, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on $k$-independence, J. Graph Theory 15 (1991), 99-107].

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On Some Conjectures Concerning Critical Independent Sets of a Graph

The Electronic Journal of Combinatorics ◽

10.37236/5580 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 5

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.

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Roman domination and 2-independence in trees

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830917500239 ◽

2017 ◽

Vol 09 (02) ◽

pp. 1750023 ◽

Cited By ~ 1

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.

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Lower Bounds for the Average Genus of a CF-graph

The Electronic Journal of Combinatorics ◽

10.37236/422 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 2

Author(s):

Yichao Chen

Keyword(s):

Lower Bound ◽

Linear Function ◽

Lower Bounds ◽

Betti Number ◽

Maximum Genus ◽

Simple Graphs ◽

Structure Theorems ◽

Average Genus

CF-graphs form a class of multigraphs that contains all simple graphs. We prove a lower bound for the average genus of a CF-graph which is a linear function of its Betti number. A lower bound for average genus in terms of the maximum genus and some structure theorems for graphs with a given average genus are also provided.

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Planar Graphs have Independence Ratio at least 3/13

The Electronic Journal of Combinatorics ◽

10.37236/5309 ◽

2016 ◽

Vol 23 (3) ◽

Cited By ~ 1

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 $\frac{3n}{13}$.

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New Bounds for the α-Indices of Graphs

Mathematics ◽

10.3390/math8101668 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1668

Author(s):

Eber Lenes ◽

Exequiel Mallea-Zepeda ◽

Jonnathan Rodríguez

Keyword(s):

Lower Bound ◽

Spectral Radius ◽

Upper Bound ◽

Adjacency Matrix ◽

Independence Number ◽

Minimal Degree ◽

Diagonal Matrix ◽

The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

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