On the sigma value and sigma range of the join of a finite number of even cycles of the same order

Abstract

Abstract Let c : V(G) → N be a vertex coloring of a simple, connected graph G. For a vertex v of G, the color sum of v, denoted by σ(ν), is the sum of the colors of the neighbors of v. If σ(ω) = σ(ν) for any two adjacent vertices u and v of G, then c is called a sigma coloring of G. The sigma chromatic number of G, denoted by σ(G), is the minimum number of colors required in a sigma coloring of G. Let max(c) be the largest color assigned to a vertex of G by a coloring c. The sigma value of G, denoted by v(G), is the minimum value of max(c) over all sigma k—colorings c of G for which ρ(G) = k. On the other hand, the sigma range of G, denoted by ρ(G), is the minimum value of max(c) over all sigma colorings c of G. In this paper, we determine the sigma value and the sigma range of the join of a finite number of even cycles of the same order. In particular, if n > 4 and n is even, then we will show that ρ(kCn ) = v(kCn) = 2 if and only if (i) k ≤ [ n 6 ] + 1 , whenever n = 0 (mod 4), and (ii) k ≤ [ n − 2 6 ] + 1 , whenever n = 2 (mod 4).