#### Document Type

Conference Proceeding

#### Publication Date

2020

#### Abstract

Let *c* : *V*(*G*) → be a coloring of the vertices in a graph *G*. For a vertex *u* in *G*, the color sum of *u*, denoted by *σ*(*u*), is the sum of the colors of the neighbors of *u*. The coloring *c* is called a sigma coloring of *G* if *σ*(*u*) ≠ *σ*(*v*) whenever *u* and *v* are adjacent vertices in *G*. The minimum number of colors that can be used in a sigma coloring of *G* is called the sigma chromatic number of *G* and is denoted by *σ*(*G*). Given two simple, connected graphs *G* and *H*, the corona of *G* and *H*, denoted by *G* ⊙ *H*, is the graph obtained by taking one copy of *G* and |*V*(*G*)| copies of *H* and where the *i*th vertex of *G* is adjacent to every vertex of the *i*th copy of *H*. In this study, we will show that for a graph *G* with |*V*(*G*)| ≥ 2, and a complete graph *Kn *of order *n*, *n* ≤ *σ*(*G* ⊙ *Kn *) ≤ max {*σ*(*G*), *n*}. In addition, let *Pn *and *Cn *denote a path and a cycle of order *n* respectively. If *m, n* ≥ 3, we will prove that *σ*(*Km *⊙ *Pn *) = 2 if and only if . If *n* is even, we show that *σ*(*Km *⊙ *Cn *) = 2 if and only if . Furthermore, in the case that *n* is odd, we show that *σ*(*Km *⊙ *Cn *) = 3 if and only if where *H*(*r, s*) denotes the number of lattice points in the convex hull of points on the plane determined by the integer parameters *r* and *s*.

#### Recommended Citation

Garciano, A., Lagura, M., & Marcelo, R. (2020). Sigma chromatic number of graph coronas involving complete graphs. Journal of Physics: Conference Series, 1538. doi:10.1088/1742-6596/1538/1/012003