2-Distance chromatic number of some graph products

Abstract

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

On the Packing Partitioning Problem on Directed Graphs

“…Ghazi et al [9] exhibited the exact formula χ 2 (G • H) = χ 2 (G)|V (H)| for all connected graphs G and H. In what follows, we show that this equality is not true as it stands. In Figure 4, we consider the lexicographic product graph C 7 • C 5 without drawing the edges (for the sake of convenience).…”

“…In the case of corona product graphs, we give a closed formula for this parameter and prove that it assumes all values given in the formula. In particular, a counterexample to a formula given in [9] concerning the 2-distance chromatic number of lexicographic product graphs is presented. Moreover, when dealing with the direct product graphs, we completely determine the injective chromatic number of direct product of two cycles by using the new tools given in this paper and some classical results in the literature.…”