Coloring, location and domination of corona graphs

TL;DR

This paper explores coloring, domination, and location properties of corona graphs, providing bounds, exact values, and formulas for various parameters related to these properties in corona graph structures.

Contribution

It introduces new bounds, exact values, and formulas for coloring and domination parameters specifically in corona graphs, advancing understanding of their structural properties.

Findings

Tight bounds for distance-2 and distance-3 chromatic numbers of corona graphs.

Exact distance-$k$ chromatic number for corona of a path and any graph.

Formulas for various domination and independence parameters in corona graphs.

Abstract

A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set $S$ of vertices of a graph $G$ is a dominating set in $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. A domination parameter of $G$ is related to those structures of a graph satisfying some domination property together with other conditions on the vertices of $G$. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-$k$ colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, throughout some relationships between the distance-$k$ chromatic number of corona graphs and the distance-$k$ chromatic number of its factors. Moreover, we give the exact value of the distance-$k$ chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating-domination number of corona graphs. We give closed formulaes for the $k$-domination number, the distance-$k$ domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.