**Authors:** Narahari Narasimha Swamy, Badekara Sooryanarayana, Akshara Prasad S. P.

**Keywords:** Sigma Coloring, open neighborhood sum, magic sigma coloring, sigma chromatic number.

### Abstract

A sigma coloring of a non-trivial connected graph $G$ is a coloring $c:V(G)\rightarrow\mathbb{N}$ such that $\sigma(u)\ne\sigma(v)$ for every two adjacent vertices $u,v\in V(G)$, where $\sigma(v)$ is the sum of the colors of the vertices in the open neighborhood $N(v)$ of $v\in V(G)$. The minimum number of colors required in a sigma coloring of a graph $G$ is called the sigma chromatic number of $G$, denoted $\sigma(G)$. A coloring $c:V(G)\rightarrow \{1, 2, \cdots, k\}$ is said to be a magic sigma coloring of $G$ if the sum of colors of all the vertices in the open neighborhood of each vertex of $G$ is the same. In this paper, we study some of the properties of magic sigma coloring of a graph. Further, we define the magic sigma chromatic number of a graph and determine it for some known families of graphs.

Narahari Narasimha Swamy

Department of Mathematics

University College of Science

Tumkur University, Tumakuru - 572103

Karnataka, India

Phone:+919739482878

E-mail:

Badekara Sooryanarayana

Department of Mathematical and Computational Studies

Dr. Ambedkar Institute of Technology, Bengaluru - 560056

Karnataka, India

Phone:+919844236450

E-mail:

Akshara Prasad S. P.

Department of Mathematics

University College of Science

Tumkur University, Tumakuru - 572103

Karnataka, India

Phone:+9190355 83206

E-mail:

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