Some Results on Modular Coloring Problems of Some Graph Operations

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DOI: https://doi.org/10.28924/2291-8639-24-2026-106
Published: Apr 7, 2026
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S. Mahalakshmi, G. Rajasekaran, Some Results on Modular Coloring Problems of Some Graph Operations, Int. J. Anal. Appl., 24 (2026), 106.

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S. Mahalakshmi, G. Rajasekaran

Abstract

Let k ≥ 2 be an integer. A modular k-coloring of a graph G is defined as a mapping from V(G) to the set Zk such that adjacent vertices may receive the same color and their color sums are distinct under modulo k. The least integer k that admits a valid modular coloring is referred to the modular chromatic number of G, represented by χmc(G). In this study, our focus is on the modular chromatic number for certain graph classes, particularly type-1 and type-2 trees and some operations of graphs and those are: cartesian product, rooted product and join graphs. Also, we provide a graph G satisfying χmc(G) = χ(G) + 1.

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