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

Authors

Agnes Garciano, Ateneo de Manila UniversityFollow
Maria Czarina T. Lagura
Reginaldo M. Marcelo, Ateneo de Manila UniversityFollow

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 (H₁, H₂) ≠ (C₆, C₆); 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.