scholarly journals Solutions to four open problems on quorum colorings of graphs

2021 ◽  
Author(s):  
Rafik Sahbi
Keyword(s):  
Maximum Cardinality ◽  
Open Problems ◽  
Coloring Number ◽  
Vertex Set ◽  
Good Characterization ◽  
Closed Neighborhood

A partition $\pi=\{V_{1},V_{2},...,V_{k}\}$ of the vertex set $V$ of a graph $G$ into $k$ color classes $V_{i},$ with $1\leq i\leq k$ is called a quorum coloring if for every vertex $v\in V,$ at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v$. The maximum cardinality of a quorum coloring of $G$ is called the quorum coloring number of $G$ and is denoted $\psi_{q}(G).$ In this paper, we give answers to four open problems stated in 2013 by Hedetniemi, Hedetniemi, Laskar and Mulder. In particular, we show that there is no good characterization of the graphs $G$ with $\psi_{q}(G)=1$ nor for those with $\psi_{q} (G)>1$ unless $\mathcal{P}\neq\mathcal{NP}\cap co-\mathcal{NP}.$ We also construct several new infinite  families of such graphs, one of which the diameter $diam(G)$ of $G$ is not bounded.

scholarly journals Secret Sharing Schemes on Sparse Homogeneous Access Structures with Rank Three

10.37236/1825 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Jaume Martí-Farré ◽  
Carles Padró
Keyword(s):  
Secret Sharing ◽  
Access Structure ◽  
Open Problems ◽  
Access Structures ◽  
The Family ◽  
Ideal Access Structures ◽  
Optimal Information ◽  
Made In ◽  
The Ideal

One of the main open problems in secret sharing is the characterization of the ideal access structures. This problem has been studied for several families of access structures with similar results. Namely, in all these families, the ideal access structures coincide with the vector space ones and, besides, the optimal information rate of a non-ideal access structure is at most $2/3$. An access structure is said to be $r$-homogeneous if there are exactly $r$ participants in every minimal qualified subset. A first approach to the characterization of the ideal $3$-homogeneous access structures is made in this paper. We show that the results in the previously studied families can not be directly generalized to this one. Nevertheless, we prove that the equivalences above apply to the family of the sparse $3$-homogeneous access structures, that is, those in which any subset of four participants contains at most two minimal qualified subsets. Besides, we give a complete description of the ideal sparse $3$-homogeneous access structures.

A study of enhanced power graphs of finite groups

2019 ◽  
Vol 19 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Samir Zahirović ◽  
Ivica Bošnjak ◽  
Rozália Madarász
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Finite Abelian Groups ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

scholarly journals Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )

2018 ◽  
Vol 6 (1) ◽  
pp. 343-356
Author(s):  
K. Arathi Bhat ◽  
G. Sudhakara
Keyword(s):  
Chromatic Number ◽  
Perfect Matching ◽  
Domination Number ◽  
Perfect Matchings ◽  
Factor Graph ◽  
Vertex Set ◽  
Graph Parameters

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

scholarly journals Efficient Algorithm for Generating Maximal L-Reflexive Trees

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 809
Author(s):  
Milica Anđelić ◽  
Dejan Živković
Keyword(s):  
Efficient Algorithm ◽  
Line Graph ◽  
Common Vertex ◽  
Line Graphs ◽  
Algebraic Numbers ◽  
Pisot Numbers ◽  
Full Characterization ◽  
Vertex Set ◽  
Second Largest Eigenvalue

The line graph of a graph G is another graph of which the vertex set corresponds to the edge set of G, and two vertices of the line graph of G are adjacent if the corresponding edges in G share a common vertex. A graph is reflexive if the second-largest eigenvalue of its adjacency matrix is no greater than 2. Reflexive graphs give combinatorial ground to generate two classes of algebraic numbers, Salem and Pisot numbers. The difficult question of identifying those graphs whose line graphs are reflexive (called L-reflexive graphs) is naturally attacked by first answering this question for trees. Even then, however, an elegant full characterization of reflexive line graphs of trees has proved to be quite formidable. In this paper, we present an efficient algorithm for the exhaustive generation of maximal L-reflexive trees.

A CHARACTERIZATION OF RECOGNIZABLE PICTURE LANGUAGES

1994 ◽  
Vol 08 (02) ◽  
pp. 501-508 ◽  
Author(s):  
KATSUSHI INOUE ◽  
ITSUO TAKANAMI
Keyword(s):  
Finite Automata ◽  
Two Dimensional ◽  
Open Problems ◽  
Nondeterministic Finite Automata ◽  
Letter Alphabet ◽  
The Family ◽  
On Line ◽  
Picture Languages

This paper first shows that REC, the family of recognizable picture languages in Giammarresi and Restivo,3 is equal to the family of picture languages accepted by two-dimensional on-line tessellation acceptors in Inoue and Nakamura.5 By using this result, we then solve open problems in Giammarresi and Restivo,3 and show that (i) REC is not closed under complementation, and (ii) REC properly contains the family of picture languages accepted by two-dimensional nondeterministic finite automata even over a one letter alphabet.

scholarly journals VERY WELL-COVERED GRAPHS OF GIRTH AT LEAST FOUR AND LOCAL MAXIMUM STABLE SET GREEDOIDS

2011 ◽  
Vol 03 (02) ◽  
pp. 245-252 ◽  
Author(s):  
VADIM E. LEVIT ◽  
EUGEN MANDRESCU
Keyword(s):  
Perfect Matching ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
Maximum Cardinality ◽  
Math Program ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Appl Math

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter Jr. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232–248], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91–101] we have shown that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163–174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414–2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89–94], respectively. In this paper we demonstrate that if G is a very well-covered graph of girth ≥4, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.

scholarly journals On(k,kn-k2-2k-1)-Choosability ofn-Vertex Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Wongsakorn Charoenpanitseri
Keyword(s):  
Proper Coloring ◽  
Open Problems

A(k,t)-list assignmentLof a graphGis a mapping which assigns a set of sizekto each vertexvofGand|⋃v∈V(G)‍L(v)|=t. A graphGis(k,t)-choosable ifGhas a proper coloringfsuch thatf(v)∈L(v)for each(k,t)-list assignmentL. In 2011, Charoenpanitseri et al. gave a characterization of(k,t)-choosability ofn-vertex graphs whent≥kn-k2-2k+1and left open problems whent≤kn-k2-2k. Recently, Ruksasakchai and Nakprasit obtain the results whent=kn-k2-2k. In this paper, we extend the results to caset=kn-k2-2k-1.

scholarly journals Hypergraph domination and strong independence

2009 ◽  
Vol 3 (2) ◽  
pp. 347-358 ◽  
Author(s):  
Bibin Jose ◽  
Zsolt Tuza
Keyword(s):  
Dominating Set ◽  
Independent Sets ◽  
Dominating Sets ◽  
Tight Bound ◽  
Open Problems ◽  
Odd Cycle ◽  
Vertex Set

We solve several conjectures and open problems from a recent paper by Acharya [2]. Some of our results are relatives of the Nordhaus-Gaddum theorem, concerning the sum of domination parameters in hypergraphs and their complements. (A dominating set in H is a vertex set D X such that, for every vertex x? X\D there exists an edge E ? E with x ? E and E?D ??.) As an example, it is shown that the tight bound ??(H)+??(H) ? n+2 holds in hypergraphs H = (X, E) of order n ? 6, where H is defined as H = (X, E) with E = {X\E | E ? E}, and ?? is the minimum total cardinality of two disjoint dominating sets. We also present some simple constructions of balanced hypergraphs, disproving conjectures of the aforementioned paper concerning strongly independent sets. (Hypergraph H is balanced if every odd cycle in H has an edge containing three vertices of the cycle; and a set S X is strongly independent if |S?E|? 1 for all E ? E.).

A characterization of rings whose unitary Cayley graphs are planar

2018 ◽  
Vol 17 (09) ◽  
pp. 1850178 ◽  
Author(s):  
Huadong Su ◽  
Yiqiang Zhou
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

Let [Formula: see text] be a ring with identity. The unitary Cayley graph of [Formula: see text] is the simple graph with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are linked by an edge if and only if [Formula: see text] is a unit of [Formula: see text]. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar.

Characterization of Aut(J2) and Aut(McL) by Their Non-commuting Graphs

2011 ◽  
Vol 18 (02) ◽  
pp. 327-332 ◽  
Author(s):  
Lingli Wang ◽  
Wujie Shi
Keyword(s):  
Abelian Group ◽  
Commuting Graph ◽  
Vertex Set

For a non-abelian group G, we associate the non-commuting graph ∇ (G) whose vertex set is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this paper, we prove that Aut (J2) and Aut (McL) are characterized by their non-commuting graphs.

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