#### Title

On the sigma chromatic number of the join of a finite number of paths and cycles

#### Document Type

Article

#### Publication Date

2019

#### Abstract

Let G">GG be a simple connected graph and c:V(G)→ℕ">c:V(G)→Nc:V(G)→ℕ a coloring of the vertices in G.">G.G. For any v∈V(G)">v∈V(G)v∈V(G), let σ(v)">σ(v)σ(v) be the sum of colors of the vertices adjacent to v">vv. Then c">cc is called a sigma coloring of G">GG if for any two adjacent vertices u,v∈V(G),σ(v)≠σ(u).">u,v∈V(G),σ(v)≠σ(u).u,v∈V(G),σ(v)≠σ(u). The minimum number of colors needed in a sigma coloring of G">GG is the sigma chromatic number of G">GG, denoted by σ(G).">σ(G).σ(G).

In this paper; we prescribe a sigma coloring of the join of paths and cycles. As a consequence; we determine the sigma chromatic number of the join of a finite number of paths and cycles. In particular; let G=Σ^{l} i=1 H_{i }where H_{i}=P_{ni }or H_{i}= C_{ni}; with 6 ≤ n ≤ 1 ≤ ... ≤ n_{l}. If n_{i+2} - n_{i }≥ 2 where 1 ≤ i ≤ l-2 and (H_{1}, H_{2}) ≠ (C_{6}, C_{6}); then σ (G) = 3 if H_{i }is an odd cycle, for some i, and σ(G) = 2 otherwise.

#### Recommended Citation

Garciano, A. D., Lagura, M. C. T., & Marcelo, R. M. (2019). On the sigma chromatic number of the join of a finite number of paths and cycles. Asian-European Journal of Mathematics, 2150019.