New bounds on the independence number of connected graphs

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].

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

10.37236/3601 ◽  
2014 ◽  
Vol 21 (1) ◽  
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 $\tau(H) \le n/4$.

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

10.37236/2646 ◽  
2013 ◽  
Vol 20 (1) ◽  
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].

scholarly journals Transversals and Independence in Linear Hypergraphs with Maximum Degree Two

10.37236/6160 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Maximum Degree ◽  
Independence Number ◽  
Independent Set ◽  
Upper Bounds ◽  
Maximum Cardinality ◽  
Common Edge ◽  
Minimum Cardinality ◽  
Uniform Hypergraphs ◽  
Transversal Number ◽  
Linear Hypergraphs

For $k \ge 2$, let $H$ be a $k$-uniform hypergraph on $n$ vertices and $m$ edges. Let $S$ be a set of vertices in a hypergraph $H$. The set $S$ is a transversal if $S$ intersects every edge of $H$, while the set $S$ is strongly independent if no two vertices in $S$ belong to a common edge. The transversal number, $\tau(H)$, of $H$ is the minimum cardinality of a transversal in $H$, and the strong independence number of $H$, $\alpha(H)$, is the maximum cardinality of a strongly independent set in $H$. The hypergraph $H$ is linear if every two distinct edges of $H$ intersect in at most one vertex. Let $\mathcal{H}_k$ be the class of all connected, linear, $k$-uniform hypergraphs with maximum degree $2$. It is known [European J. Combin. 36 (2014), 231–236] that if $H \in \mathcal{H}_k$, then $(k+1)\tau(H) \le n+m$, and there are only two hypergraphs that achieve equality in the bound. In this paper, we prove a much more powerful result, and establish tight upper bounds on $\tau(H)$ and tight lower bounds on $\alpha(H)$ that are achieved for  infinite families of hypergraphs. More precisely, if $k \ge 3$ is odd and $H \in \mathcal{H}_k$ has $n$ vertices and $m$ edges, then we prove that $k(k^2 - 3)\tau(H) \le (k-2)(k+1)n + (k - 1)^2m + k-1$ and $k(k^2 - 3)\alpha(H) \ge  (k^2 + k - 4)n  - (k-1)^2 m - (k-1)$. Similar bounds are proven in the case when $k \ge 2$ is even.

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.

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

scholarly journals The edge geodetic self decomposition number of a graph

2020 ◽  
Author(s):  
John J ◽  
Stalin D
Keyword(s):  
Connected Graph ◽  
Maximum Cardinality ◽  
Connected Graphs ◽  
Edge Disjoint ◽  
General Properties ◽  
Geodetic Number ◽  
Simple Connected Graph ◽  
Decomposition Number

Let  G = (V, E)  be a simple connected  graph  of order  p and  size q.  A decomposition  of a graph  G is a collection  π  of edge-disjoint sub graphs  G1, G2, ..., Gn  of G such  that every  edge of G belongs to exactly  one Gi , (1 ≤ i ≤ n) . The decomposition  π = {G1, G2, ....Gn } of a connected  graph  G is said to be an edge geodetic self decomposi- tion  if ge (Gi ) = ge (G), (1 ≤ i ≤ n).The maximum  cardinality of π is called the edge geodetic self decomposition  number of G and is denoted by πsge (G), where ge (G) is the edge geodetic number  of G.  Some general properties   satisfied  by  this  concept  are  studied.    Connected  graphs which are edge geodetic self decomposable  are characterized.

Component factors of the Cartesian product of graphs

2016 ◽  
Vol 09 (02) ◽  
pp. 1650041
Author(s):  
M. R. Chithra ◽  
A. Vijayakumar
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Cartesian Product ◽  
Induced Subgraph ◽  
Connected Graphs ◽  
Cartesian Product Of Graphs ◽  
Spanning Subgraph ◽  
Component Factor ◽  
Product Of Graphs

Let [Formula: see text] be a family of connected graphs. A spanning subgraph [Formula: see text] of [Formula: see text] is called an [Formula: see text]-factor (component factor) of [Formula: see text] if each component of [Formula: see text] is in [Formula: see text]. In this paper, we study the component factors of the Cartesian product of graphs. Here, we take [Formula: see text] and show that every connected graph [Formula: see text] has a [Formula: see text]-factor where [Formula: see text] and [Formula: see text] is the maximum degree of an induced subgraph [Formula: see text] in [Formula: see text] or [Formula: see text]. Also, we characterize graphs [Formula: see text] having a [Formula: see text]-factor.

Outer-connected independence in graphs

2015 ◽  
Vol 07 (03) ◽  
pp. 1550039
Author(s):  
I. Sahul Hamid ◽  
R. Gnanaprakasam ◽  
M. Fatima Mary
Keyword(s):  
Independence Number ◽  
Independent Set ◽  
Proper Subset ◽  
Maximum Cardinality

A set S ⊆ V(G) is an independent set if no two vertices of S are adjacent. An independent set S such that 〈V - S〉 is connected is called an outer-connected independent set(oci-set). An oci-set is maximal if it is not a proper subset of any oci-set. The minimum and maximum cardinality of a maximal oci-set are called respectively the outer-connected independence number and the upper outer-connected independence number. This paper initiates a study of these parameters.

Longest Cycles in 2-Connected Graphs with Prescribed Maximum Degree

1980 ◽  
Vol 32 (6) ◽  
pp. 1325-1332 ◽  
Author(s):  
J. A. Bondy ◽  
R. C. Entringer
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Simple Graph ◽  
General Context ◽  
Connected Graphs ◽  
Longest Cycles ◽  
Image Position ◽  
The Relationship

The relationship between the lengths of cycles in a graph and the degrees of its vertices was first studied in a general context by G. A. Dirac. In [5], he proved that every 2-connected simple graph on n vertices with minimum degree d contains a cycle of length at least min{2d, n};. Dirac's theorem was subsequently strengthened in various directions in [7], [6], [13], [12], [2], [1], [11], [8], [14], [15] and [16].Our aim here is to investigate another aspect of this relationship, namely how the lengths of the cycles in a 2-connected graph depend on the maximum degree. Let us denote by ƒ(n, d) the largest integer k such that every 2-connected simple graph on n vertices with maximum degree d contains a cycle of length at least k. We prove in Section 2 that, for d ≧ 3 and n ≧ d + 2,

scholarly journals Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Gábor Bacsó ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Connected Graph ◽  
Maximum Degree ◽  
Free Graph ◽  
Polynomial Time Algorithms ◽  
Odd Cycle ◽  
Vertex Set ◽  
International Audience

Graphs and Algorithms International audience A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

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