On k-rainbow domination in middle graphs

Kijung Kim

doi:10.1051/ro/2021163

Abstract

Let G be a finite simple graph with vertex set V(G) and edge set E(G). A function f : V(G) → P({1,2,…,k}) is a k-rainbow dominating function on G if for each vertex v∈V(G) for which f(v) = ∅, it holds that ⋃u∈N(v) f(u) = {1,2,…,k}. The weight of a k-rainbow dominating function is the value ∑v∈V(G)|f(v)|. The k-rainbow domination number γrk (G) is the minimum weight of a k-rainbow dominating function on G. In this paper, we initiate the study of k-rainbow domination numbers in middle graphs. We define the concept of a middle k-rainbow dominating function, obtain some bounds related to it and determine the middle 3-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle 3-rainbow domination number of trees in terms of the matching number. In addition, we determine the 3-rainbow domatic number for the middle graph of paths and cycles.

Highlights

Let G = (V, E) be a connected undirected graph with the vertex set V = V (G) and edge set E = E(G)
The maximum degree and minimum degree of G are denoted by ∆(G) and δ(G), respectively
We provide upper and lower bounds for the middle 3-rainbow domination number of trees in terms of the matching number

Summary

Introduction

It is worthwhile to determine the k-rainbow domination numbers of some classes of graphs. For k ≥ 3, it is more difficult to determine the k-rainbow domination number of a graph. In [10], Shao et al determined the 3-rainbow domination numbers of paths, cycles and generalized Petersen graphs P (n, 1). To study k-rainbow domination numbers in the class of middle graphs, we define the following concept. In [9], only 2-rainbow domination numbers of the middle graphs were considered. The maximum number of functions in a k-rainbow dominating family on G is the k-rainbow domatic number of G, denoted by drk(G). We determine the 3-rainbow domatic number for the middle graph of paths and cycles.

General bounds of the middle k-rainbow domination number

Lower and upper bounds for trees

The 3-rainbow domatic number for the middle graph of paths and cycles