• PROPER MODULAR CHROMATIC NUMBER OF CIRCULAR HALIN GRAPHS OF LEVEL TWO

Abstract

For a connected graph G, let c: V(G) → ℤ_k (k ≥ 2) be a vertex coloring of G. The color sum (v) of a vertex v of G is defined as the sum in ℤ_k of the colors of the vertices in N(v), that is 𝜎(v) =  (mod k). The coloring c is called a modular k-coloring of G if 𝜎(x) ≠ 𝜎(y) in ℤ_kfor all pairs of adjacent vertices x, y G. The modular chromatic number or simply the mc-number of G is the minimum k for which G has a modular k-coloring.