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=Σl i=1 Hi where Hi=Pni or Hi= Cni; with 6 ≤ n ≤ 1 ≤ ... ≤ nl. If ni+2 - ni ≥ 2 where 1 ≤ i ≤ l-2 and (H1, H2) ≠ (C6, C6); then σ (G) = 3 if Hi is an odd cycle, for some i, and σ(G) = 2 otherwise.

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