Vellore Institute of Technology (VIT) is a private university located in&nbsp;Tamil Nadu, India. Founded in 1984, as Vellore Engineering College, the institution offers 20 undergraduate, 34 postgraduate, four integrated and four research programs. It has campuses in Vellore, Amravati, Bhopal and Chennai.

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b-Coloring of G is a coloring which is proper such that in each color class there exists a vertex which is called as representative vertex that has at least one neighbor in each of the remaining color classes. The highest positive integer k such that the k-colors can be used to color the vertices of G along with b-coloring is the b-chromatic number of G, denoted by b(G). For a given graph G with n vertices, G ∗ is constructed (Jeeva et al., Indian J Math 59(2):255–261, 2017). In the research paper, we find out the b-chromatic number of Mycielskian, splitting, shadow, middle, and total graph of G ∗ . © Springer Nature Switzerland AG 2019.

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Department of Mathematics

School of Advanced Sciences

Vellore Campus

Trends in Mathematics

The b-chromatic number of some standard graphs

Department of Information Security

School of Computer Science and Engineering

Electrical

School of Electrical Engineering

Chennai Campus

Preface

Given G, b - coloring is a proper k coloring of G in which each and every color class has at least one b - vertex that has a neighbour in other k - 1 color classes. The largest integer k is the b - chromatic number b(G) for which G having a b - coloring using k colors. In this paper, we constructed some family of graphs and found its b - chromatic number. © Published under licence by IOP Publishing Ltd.

VIT Unitversity

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IOP Conference Series: Materials Science and Engineering

The b-chromatic number of some special families of graphs

An (a,d)-edge antimagic total labeling of a (p,Â q)-graph G is bijection f: V ∪ E → {1,2,3,...,p+q} with the property that the edge-weights w(uv)= f(u)+ f(v)+ f(uv) where uv ∈ E(G) form an arithmetic progression a, a+d,..., a+(q-1)d, where a &gt;0 and d ≥ 0 are two fixed integers. If such a labeling exists, then G is called an (a,d)-edge antimagic total graph. If further the vertex labels are the integers {1,2,3,...,p}, then f is called a super (a,d)-edge antimagic total labeling of G ((a, d)-SEAMT labeling) and a graph which admits such a labeling is called a super (a,d)-edge antimagic total graph ((a, d)-SEAMT graph). If d=0, then the graph G is called a super edge-magic graph. In this paper we investigate the existence of super (a, 3)-edge antimagic total labelings for union of two stars. © 2017, Springer International Publishing AG.

Theoretical Computer Science and Discrete Mathematics Lecture Notes in Computer Science

Super (a, 3)-edge Antimagic Total Labeling for Union of Two Stars

Given a graph G, b-coloring is a proper fc-coloring of G in which every color class has at least one vertex that has a neighbour in each of the other color classes. Such a vertex is called a b-vertex. A set S0 ⊆ V is called a b-system if all the vertices in S0 are b-vertices that belong to different color classes. The b-chromatic number is the largest integer κ such that G admits a b- coloring with κ colors. AJ greater b-chromatic graph is a graph G such that the achromatic number of G is greater than the b-chromatic number of every graph obtained from G by identifying two adjacent vertices, denoted by bh,-chromatic. In this paper, for any given graph G we constructed the graphs G∗, G+ and G∗+. It is proved thatjG∗, G+ axe bh-chromatic graph but G∗+ is noi bh-chromatic graph. Further, we proved that the b-chromatic number of G∗+ is neither equal nor less than the b-chromatic number of every graph obtained from G∗+ by identifying any two adjacent vertices. © 2017 Allahabad Mathematical Society.

Indian Journal of Mathematics

Families of greater achromatic graphs

Journal of the Indonesian Mathematical Society

ANTIMAGIC LABELING OF DIGRAPHS

A b-coloring is a proper k-coloring of the vertices of a graph such that each color class has a vertex that is adjacent to a vertex of every other color class. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph G is said to be stable b-chromatic graph if b(G.uv) = b(G) for every u,v ε V (G) with u is adjacent to v. In this paper we obtain some basic properties of bs-chromatic graphs. © 2016 Academic Publications, Ltd.

International Journal of Pure and Applied Mathematics

The stable b-chromatic graphs

An (a, d)-edge-antimagic total labeling of a graph G = (V, E) with p vertices and q edges is a bijection f: V ∪ E → {1,2,3,...,p + q} such that all the edge-weights ω(uv) = f(u) + f(v) + f(uv); uv ∈ E(G), form an arithmetic progression starting from a and having common difference d. An (a, d)-edge-antimagic total labeling is called a super (a, d)-edge-antimagic total labeling ((a, d)-SEAMT labeling) if f(V(G)) = {1,2,3,...,p}. In this paper we investigate the existence of (a, 3)-SEAMT labelings for union of two stars.

Utilitas Mathematica

Super (a, 3)-edge-antimagic total labelings for union of two stars

An (a, d)-edge-antimagic total labeling of a graph G with p vertices and q edges is a bijection f from the set of all vertices and edges to the set of positive integers {1, 2, 3, ..., p + q} such that all the edge-weights w(uv)=f(u)+f(v)+f(uv); uv∈E(G), form an arithmetic progression starting from a and having common difference d. An (a, d)-edge-antimagic total labeling is called a super (a, d)-edge-antimagic total labeling ((a, d)-SEAMT labeling) if f(V(G)) = {1, 2, 3, ..., p}. In this paper we investigate the existence of super (a, d)-edge antimagic total labeling for friendship graphs and generalized friendship graphs. © 2015 .

Electronic Notes in Discrete Mathematics

Super (a, d)-edge antimagic total labelings of friendship and generalized friendship graphs

An (a, d)-edge-antimagic total labeling of a graph G with p vertices and q edges is a bijection f from the set of all vertices and edges to the set of positive integers {1,2,3, ...,p + q} such that all the edge-weights w(uv) = f(u) + f(v) + f(uv);uv ε E(G), form an arithmetic progression starting from a and having common difference d. An (a, d)-edge-antimagic total labeling is called a super (a, d)-edge-antimagic total labeling ((a, d)-SEAMT labeling) if f(V(G)) = {l,2,3,...,p}. The graph Fn consisting of n triangles with a common vertex is called the friendship graph. The generalized friendship graph Fm1,m2.....mn consists of n cycles of orders m1 ≤ m2 ≤,...,≤ mn having a common vertex. In this paper we prove that the friendship graph F16 does not admit a (a, 2)-SEAMT labeling. We also investigate the existence of (a, d)-SEAMT labeling for several classes of generalized friendship graphs.

Journal of Combinatorial Mathematics and Combinatorial Computing

Super (a, d)-edge-antimagic total labelings of generalized friendship graphs

An (a, d)-edge-antimagic total labeling of a graph G with p vertices and q edges is a bijection f from the set of all vertices and edges to the set of positive integers {1, 2, 3,., p + q} such that all the edge-weights w(uν) = f(u) + f(ν) + f(uν); uν ∈ E(G), form an arithmetic progression starting from a and having common difference d. An (a, d)-edgeantimagic total labeling is called a super (a, d)-edge-antimagic total labeling ((a, d)-SEAMT labeling) if f(V (G)) = {1, 2, 3,., p}. In this paper we investigate the existence of super (a, d)-edge antimagic total labeling for friendship graphs and generalized friendship graphs.