Color hypergroup and join space obtained by the vertex coloring of a graph

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6501-6513 ◽  
Author(s):  
M. Golmohamadian ◽  
M.M. Zahedi
Keyword(s):  
Graph Theory ◽  
Vertex Coloring ◽  
Join Space ◽  
Vertex Set

In this paper, some new classes of hyperoperations, extracting from the vertex coloring of a graph, are presented. By these hyperoperations, we define a color hypergroup and a color join space on the vertex set of a graph. Also we give some examples to clarify these structures. Finally, we investigate the connection between the color join space and Graph Theory.

scholarly journals Order divisor graphs of finite groups

2018 ◽  
Vol 26 (3) ◽  
pp. 29-40
Author(s):  
S. U. Rehman ◽  
A. Q. Baig ◽  
M. Imran ◽  
Z. U. Khan
Keyword(s):  
Graph Theory ◽  
Finite Groups ◽  
Algebraic Graph Theory ◽  
Vertex Set ◽  
Productive Area

AbstractThe interplay between groups and graphs have been the most famous and productive area of algebraic graph theory. In this paper, we introduce and study the graphs whose vertex set is group G such that two distinct vertices a and b having di erent orders are adjacent provided that o(a) divides o(b) or o(b) divides o(a).

scholarly journals Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yajing Wang ◽  
Yubin Gao
Keyword(s):  
Graph Theory ◽  
Spectral Radius ◽  
Adjacency Matrix ◽  
Upper Bounds ◽  
Simple Graph ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Spectral Graph ◽  
Vertex Set

Spectral graph theory plays an important role in engineering. Let G be a simple graph of order n with vertex set V=v1,v2,…,vn. For vi∈V, the degree of the vertex vi, denoted by di, is the number of the vertices adjacent to vi. The arithmetic-geometric adjacency matrix AagG of G is defined as the n×n matrix whose i,j entry is equal to di+dj/2didj if the vertices vi and vj are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus–Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.

Some algebraic hyperstructures related to zero forcing sets and forcing digraphs

2019 ◽  
Vol 18 (10) ◽  
pp. 1950192
Author(s):  
M. Golmohamadian ◽  
M. M. Zahedi ◽  
N. Soltankhah
Keyword(s):  
Graph Theory ◽  
Zero Forcing ◽  
Vertex Set ◽  
The Relationship ◽  
Forcing Set ◽  
Zero Forcing Set

A zero forcing set is a new concept in Graph Theory which was introduced in recent years. In this paper, we investigate the relationship between zero forcing sets and algebraic hyperstructures. To this end, we present some new definitions by considering a zero forcing process on a graph [Formula: see text]. These definitions help us analyze the zero forcing process better and construct various hypergroups and join spaces on the vertex set of graph [Formula: see text]. Finally, we give some examples to clarify these hyperstructures.

scholarly journals Edge stability in secure graph domination

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Anton Pierre Burger ◽  
Alewyn Petrus Villiers ◽  
Jan Harm Vuuren
Keyword(s):  
Graph Theory ◽  
Dominating Set ◽  
Domination Number ◽  
Arbitrary Subset ◽  
Graph Domination ◽  
Vertex Set ◽  
International Audience ◽  
Edge Stability ◽  
Secure Domination

Graph Theory International audience A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

On the complexity of alpha conversion

2007 ◽  
Vol 72 (4) ◽  
pp. 1197-1203
Author(s):  
Rick Statman
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Vertex Coloring ◽  
Feedback Vertex Set ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
The Way

AbstractWe consider three problems concerning alpha conversion of closed terms (combinators).(1) Given a combinator M find the an alpha convert of M with a smallest number of distinct variables.(2) Given two alpha convertible combinators M and N find a shortest alpha conversion of M to N.(3) Given two alpha convertible combinators M and N find an alpha conversion of M to N which uses the smallest number of variables possible along the way.We obtain the following results.(1) There is a polynomial time algorithm for solving problem (1). It is reducible to vertex coloring of chordal graphs.(2) Problem (2) is co-NP complete (in recognition form). The general feedback vertex set problem for digraphs is reducible to problem (2).(3) At most one variable besides those occurring in both M and N is necessary. This appears to be the folklore but the proof is not familiar. A polynomial time algorithm for the alpha conversion of M to N using at most one extra variable is given.There is a tradeoff between solutions to problem (2) and problem (3) which we do not fully understand.

scholarly journals On Feasible Sets of Mixed Hypergraphs

10.37236/1772 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Daniel Král
Keyword(s):  
Vertex Coloring ◽  
Np Hard ◽  
Finite Sets ◽  
Feasible Set ◽  
Feasible Sets ◽  
Vertex Set ◽  
Positive Integers ◽  
Mixed Hypergraph ◽  
Families Of Subsets

A mixed hypergraph $H$ is a triple $(V,{\cal C},{\cal D})$ where $V$ is the vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, called ${\cal C}$-edges and ${\cal D}$-edges. A vertex coloring of $H$ is proper if each ${\cal C}$-edge contains two vertices with the same color and each ${\cal D}$-edge contains two vertices with different colors. The spectrum of $H$ is a vector $(r_1,\ldots,r_m)$ such that there exist exactly $r_i$ different colorings using exactly $i$ colors, $r_m\ge 1$ and there is no coloring using more than $m$ colors. The feasible set of $H$ is the set of all $i$'s such that $r_i\ne 0$. We construct a mixed hypergraph with $O(\sum_i\log r_i)$ vertices whose spectrum is equal to $(r_1,\ldots,r_m)$ for each vector of non-negative integers with $r_1=0$. We further prove that for any fixed finite sets of positive integers $A_1\subset A_2$ ($1\notin A_2$), it is NP-hard to decide whether the feasible set of a given mixed hypergraph is equal to $A_2$ even if it is promised that it is either $A_1$ or $A_2$. This fact has several interesting corollaries, e.g., that deciding whether a feasible set of a mixed hypergraph is gap-free is both NP-hard and coNP-hard.

scholarly journals On the 1-2-3-conjecture

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Akbar Davoodi ◽  
Behnaz Omoomi
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Vertex Coloring ◽  
Cartesian Product Of Graphs ◽  
Edge Weighting ◽  
International Audience ◽  
Product Of Graphs ◽  
Proper Vertex

Graph Theory International audience A k-edge-weighting of a graph G is a function w:E(G)→{1,…,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v∈V(G), c(v)=∑e∼vw(e). If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge-weighting (VC k-EW). Karoński et al. (J. Combin. Theory Ser. B, 91 (2004) 151 13;157) conjectured that every graph admits a VC 3-EW. This conjecture is known as the 1-2-3-conjecture. In this paper, first, we study the vertex-coloring edge-weighting of the Cartesian product of graphs. We prove that if the 1-2-3-conjecture holds for two graphs G and H, then it also holds for G□H. Also we prove that the Cartesian product of connected bipartite graphs admits a VC 2-EW. Moreover, we present several sufficient conditions for a graph to admit a VC 2-EW. Finally, we explore some bipartite graphs which do not admit a VC 2-EW.

Lower bounds on approximating some variations of vertex coloring problem over restricted graph classes

2020 ◽  
Vol 12 (06) ◽  
pp. 2050086
Author(s):  
Sayani Das ◽  
Sounaka Mishra
Keyword(s):  
Vertex Coloring ◽  
Restricted Class ◽  
Coloring Problem ◽  
Minimization Problems ◽  
Graph Classes ◽  
Color Class ◽  
Vertex Set ◽  
Vertex Coloring Problem ◽  
Text Color ◽  
Dominator Coloring

A [Formula: see text] vertex coloring of a graph [Formula: see text] partitions the vertex set into [Formula: see text] color classes (or independent sets). In minimum vertex coloring problem, the aim is to minimize the number of colors used in a given graph. Here, we consider three variations of vertex coloring problem in which (i) each vertex in [Formula: see text] dominates a color class, (ii) each color class is dominated by a vertex and (iii) each vertex is dominating a color class and each color class is dominated by a vertex. These minimization problems are known as Min-Dominator-Coloring, Min-CD-Coloring and Min-Domination-Coloring, respectively. In this paper, we present approximation hardness results for these problems for some restricted class of graphs.

scholarly journals On Planar Mixed Hypergraphs

10.37236/1579 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Daniel Král'
Keyword(s):  
Chromatic Number ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Vertex Coloring ◽  
Incidence Graph ◽  
Feasible Set ◽  
Vertex Set ◽  
Bridgeless Graph ◽  
Np Complete ◽  
Mixed Hypergraph

A mixed hypergraph $H$ is a triple $(V,{\cal C},{\cal D})$ where $V$ is its vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, ${\cal C}$–edges and ${\cal D}$–edges. A mixed hypergraph is a bihypergraph iff ${\cal C}={\cal D}$. A hypergraph is planar if its bipartite incidence graph is planar. A vertex coloring of $H$ is proper if each ${\cal C}$–edge contains two vertices with the same color and each ${\cal D}$–edge contains two vertices with different colors. The set of all $k$'s for which there exists a proper coloring using exactly $k$ colors is the feasible set of $H$; the feasible set is called gap-free if it is an interval. The minimum (maximum) number of the feasible set is called a lower (upper) chromatic number. We prove that the feasible set of any planar mixed hypergraph without edges of size two and with an edge of size at least four is gap-free. We further prove that a planar mixed hypergraph with at most two ${\cal D}$–edges of size two is two-colorable. We describe a polynomial-time algorithm to decide whether the lower chromatic number of a planar mixed hypergraph equals two. We prove that it is NP-complete to find the upper chromatic number of a mixed hypergraph even for 3-uniform planar bihypergraphs. In order to prove the latter statement, we prove that it is NP-complete to determine whether a planar 3-regular bridgeless graph contains a $2$-factor with at least a given number of components.

scholarly journals On the Cartesian product of of an arbitrarily partitionable graph and a traceable graph

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Olivier Baudon ◽  
Julien Bensmail ◽  
Rafał Kalinowski ◽  
Antoni Marczyk ◽  
Jakub Przybyło ◽  
...  
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Vertex Set ◽  
International Audience ◽  
Positive Integers

Graph Theory International audience A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence τ=(n1,\textellipsis,nk) of positive integers that sum up to n, there exists a partition (V1,\textellipsis,Vk) of the vertex set V(G) such that each set Vi induces a connected subgraph of order ni. A graph G is called AP+1 if, given a vertex u∈V(G) and an index q∈ {1,\textellipsis,k}, such a partition exists with u∈Vq. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G□ H is AP+1. We also prove that G□H is AP, whenever G and H are AP and the order of one of them is not greater than four.

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