scholarly journals On the Distance Pattern Distinguishing Number of a Graph

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Sona Jose ◽  
Germina K. Augustine
Keyword(s):  
Nonempty Subset ◽  
Simple Graph ◽  
Distinguishing Number ◽  
Minimum Number

LetG=(V,E)be a connected simple graph and letMbe a nonempty subset ofV. TheM-distance pattern of a vertexuinGis the set of all distances fromuto the vertices inM. If the distance patterns of all vertices inVare distinct, then the setMis a distance pattern distinguishing set ofG. A graphGwith a distance pattern distinguishing set is called a distance pattern distinguishing graph. Minimum number of vertices in a distance pattern distinguishing set is called distance pattern distinguishing number of a graph. This paper initiates a study on the problem of finding distance pattern distinguishing number of a graph and gives bounds for distance pattern distinguishing number. Further, this paper provides an algorithm to determine whether a graph is a distance pattern distinguishing graph or not and hence to determine the distance pattern distinguishing number of that graph.

2-distance vertex-distinguishing total coloring of graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850018
Author(s):  
Yafang Hu ◽  
Weifan Wang
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Number Formula ◽  
Proper Total Coloring ◽  
Distance Formula

A [Formula: see text]-distance vertex-distinguishing total coloring of a graph [Formula: see text] is a proper total coloring of [Formula: see text] such that any pair of vertices at distance [Formula: see text] have distinct sets of colors. The [Formula: see text]-distance vertex-distinguishing total chromatic number [Formula: see text] of [Formula: see text] is the minimum number of colors needed for a [Formula: see text]-distance vertex-distinguishing total coloring of [Formula: see text]. In this paper, we determine the [Formula: see text]-distance vertex-distinguishing total chromatic number of some graphs such as paths, cycles, wheels, trees, unicycle graphs, [Formula: see text], and [Formula: see text]. We conjecture that every simple graph [Formula: see text] with maximum degree [Formula: see text] satisfies [Formula: see text].

Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions

2003 ◽  
Vol 12 (5-6) ◽  
pp. 477-494 ◽  
Author(s):  
Noga Alon ◽  
Michael Krivelevich ◽  
Benny Sudakov
Keyword(s):  
Positive Constant ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Ramsey Number ◽  
Absolute Positive Constant ◽  
Turán Number ◽  
Minimum Number ◽  
Ramsey Type ◽  
General Graphs ◽  
Refined Version

For a graph H and an integer n, the Turán number is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is r-degenerate if every one of its subgraphs contains a vertex of degree at most r. We prove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, . This is tight for all values of r and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant c such that, for every fixed bipartite r-degenerate graph H, This is motivated by a conjecture of Erdős that asserts that, for every such H, For two graphs G and H, the Ramsey number is the minimum number n such that, in any colouring of the edges of the complete graph on n vertices by red and blue, there is either a red copy of G or a blue copy of H. Erdős conjectured that there is an absolute constant c such that, for any graph G with m edges, . Here we prove this conjecture for bipartite graphs G, and prove that for general graphs G with m edges, for some absolute positive constant c.These results and some related ones are derived from a simple and yet surprisingly powerful lemma, proved, using probabilistic techniques, at the beginning of the paper. This lemma is a refined version of earlier results proved and applied by various researchers including Rödl, Kostochka, Gowers and Sudakov.

scholarly journals Families of graphs with maximum nullity equal to zero forcing number

2018 ◽  
Vol 6 (1) ◽  
pp. 56-67
Author(s):  
Joseph S. Alameda ◽  
Emelie Curl ◽  
Armando Grez ◽  
Leslie Hogben ◽  
O’Neill Kingston ◽  
...  
Keyword(s):  
Color Change ◽  
Simple Graph ◽  
Zero Forcing ◽  
Generalized Petersen Graphs ◽  
Zero Forcing Number ◽  
Regularity Properties ◽  
Maximum Nullity ◽  
Forcing Number ◽  
Minimum Number ◽  
Petersen Graphs

Abstract The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i =6 j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartite 3- semiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to find bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen graphs.

scholarly journals On the Connected Safe Number of Some Classes of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Rakib Iqbal ◽  
Muhammad Shoaib Sardar ◽  
Dalal Alrowaili ◽  
Sohail Zafar ◽  
Imran Siddique
Keyword(s):  
Connected Graph ◽  
Nonempty Subset ◽  
Simple Graph ◽  
Minimal Cardinality ◽  
Connected Component ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Wheel Graphs ◽  
Safe Set

For a connected simple graph G , a nonempty subset S of V G is a connected safe set if the induced subgraph G S is connected and the inequality S ≥ D satisfies for each connected component D of G∖S whenever an edge of G exists between S and D . A connected safe set of a connected graph G with minimum cardinality is called the minimum connected safe set and that minimum cardinality is called the connected safe numbers. We study connected safe sets with minimal cardinality of the ladder, sunlet, and wheel graphs.

scholarly journals A Performance on Harmonious Coloring of Barbell Graph

2019 ◽  
Vol 9 (1) ◽  
pp. 1583-1586
Keyword(s):  
Chromatic Number ◽  
Simple Graph ◽  
Minimum Number ◽  
A Performance

Given simple graph 𝑮, a harmonious chromatic number 𝝌𝒉(𝑮) is the minimum number of colors used in a graph such that no two adjacent vertices receives the same color and each combination of color seems together on atmost one edge. In this article we have determined the harmonious chromatic number of barbell and central graph of barbell graph.Over all we have given an algorithm to calculate the harmonious chromatic number of Barbell graph by depicting a quadratic time.

scholarly journals The conjugacy class graphs of non-abelian 3-groups

2020 ◽  
Vol 16 (3) ◽  
pp. 297-299
Author(s):  
Athirah Zulkarnain ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Conjugacy Class ◽  
Chromatic Number ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
Prime Power Order ◽  
Complete Graphs ◽  
Dominating Number ◽  
Minimum Number ◽  
Vertex Set ◽  
Class Graph

A graph is formed by a pair of vertices and edges. It can be related to groups by using the groups’ properties for its vertices and edges. The set of vertices of the graph comprises the elements or sets from the group while the set of edges of the graph is the properties and condition for the graph. A conjugacy class of an element  is the set of elements that are conjugated with . Any element of a group , labelled as , is conjugated to  if it satisfies  for some elements  in  with its inverse . A conjugacy class graph of a group   is defined when its vertex set is the set of non-central conjugacy classes of  . Two distinct vertices   and   are connected by an edge if and only if their cardinalities are not co-prime, which means that the order of the conjugacy classes of  and  have common factors. Meanwhile, a simple graph is the graph that contains no loop and no multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is adjacent. Moreover, a  -group is the group with prime power order. In this paper, the conjugacy class graphs for some non-abelian 3-groups are determined by using the group’s presentations and the definition of conjugacy class graph. There are two classifications of the non-abelian 3-groups which are used in this research. In addition, some properties of the conjugacy class graph such as the chromatic number, the dominating number, and the diameter are computed. A chromatic number is the minimum number of vertices that have the same colours where the adjacent vertices have distinct colours. Besides, a dominating number is the minimum number of vertices that is required to connect all the vertices while a diameter is the longest path between any two vertices. As a result of this research, the conjugacy class graphs of these groups are found to be complete graphs with chromatic number, dominating number and diameter that are equal to eight, one and one, respectively.

The restrained k-rainbow reinforcement number of graphs

2020 ◽  
pp. 2150026
Author(s):  
Saeed Shaebani ◽  
Saeed Kosari ◽  
Leila Asgharsharghi
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
The Family ◽  
Rainbow Domination ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
General Graphs

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple graph. A restrained [Formula: see text]-rainbow dominating function (R[Formula: see text]RDF) of [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the family of all subsets of [Formula: see text], such that every vertex [Formula: see text] with [Formula: see text] satisfies both of the conditions [Formula: see text] and [Formula: see text] simultaneously, where [Formula: see text] denotes the open neighborhood of [Formula: see text]. The weight of an R[Formula: see text]RDF is the value [Formula: see text]. The restrained[Formula: see text]-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an R[Formula: see text]RDF of [Formula: see text]. The restrained[Formula: see text]-rainbow reinforcement number [Formula: see text] of [Formula: see text], is defined to be the minimum number of edges that must be added to [Formula: see text] in order to decrease the restrained [Formula: see text]-rainbow domination number. In this paper, we determine the restrained [Formula: see text]-rainbow reinforcement number of some special classes of graphs. Also, we present some bounds on the restrained [Formula: see text]-rainbow reinforcement number of general graphs.

Clique doubly connected domination in the join and lexicographic product of graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050066
Author(s):  
Enrico L. Enriquez ◽  
Albert D. Ngujo
Keyword(s):  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Connected Dominating Set ◽  
Lexicographic Product ◽  
Connected Domination Number ◽  
Vertex Set ◽  
Product Of Graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] is a doubly connected dominating set if it is dominating and both [Formula: see text] and [Formula: see text] are connected. A nonempty subset [Formula: see text] of the vertex set [Formula: see text] is a clique in [Formula: see text] if the graph [Formula: see text] induced by [Formula: see text] is complete. A clique dominating set [Formula: see text] of [Formula: see text] is a clique doubly connected dominating set if [Formula: see text] is a doubly connected dominating set of [Formula: see text]. The clique doubly connected domination number of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of a clique doubly connected dominating set [Formula: see text] of [Formula: see text]. In this paper, we give the characterization of the clique doubly connected dominating set and the clique doubly connected domination number in the join (and lexicographic product) of two graphs.

Total coloring conjecture for vertex, edge and neighborhood corona products of graphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950014
Author(s):  
Radhakrishnan Vignesh ◽  
Jayabalan Geetha ◽  
Kanagasabapathi Somasundaram
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Tight Bound ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Total Coloring Conjecture

A total coloring of a graph [Formula: see text] is an assignment of colors to the elements of the graph [Formula: see text] such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any simple graph [Formula: see text], [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. In this paper, we prove the tight bound of the total coloring conjecture for the three types of corona products (vertex, edge and neighborhood) of graphs.

scholarly journals Distinguishing Numbers for Graphs and Groups

10.37236/1816 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Julianna Tymoczko
Keyword(s):  
Group Action ◽  
General Problem ◽  
Group Actions ◽  
Graph Automorphism ◽  
Arbitrary Group ◽  
Distinguishing Number ◽  
Minimum Number ◽  
Trivial Graph ◽  
Open Question

A graph $G$ is distinguished if its vertices are labelled by a map $\phi: V(G) \longrightarrow \{1,2,\ldots, k\}$ so that no non-trivial graph automorphism preserves $\phi$. The distinguishing number of $G$ is the minimum number $k$ necessary for $\phi$ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of $\Gamma$ on a set $X$. A labelling $\phi: X \longrightarrow \{1,2,\ldots,k\}$ is distinguishing if no element of $\Gamma$ preserves $\phi$ except those which fix each element of $X$. The distinguishing number of the group action on $X$ is the minimum $k$ needed for $\phi$ to distinguish the group action. We show that distinguishing group actions is a more general problem than distinguishing graphs. We completely characterize actions of $S_n$ on a set with distinguishing number $n$, answering an open question of Albertson and Collins.

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