A graph labeling problem is an assignment of labels to the vertices or edges (or both) of a graph G satisfying some mathematical condition. Radio Mean Labeling, a vertex-labeling of graphs with non-negative integers has a significant application in the study of problems related to radio channel assignment. The maximum label used in a radio mean labeling is called its span, and the lowest possible span of a radio mean labeling is called the radio mean number of a graph. In this paper, we obtain the radio mean number of paths and total graph of paths.
In this article, we point out the flaws in Theorem 1 of the article "Badr, Elsayed, et al. "An integer linear programming model for solving radio mean labeling problem." IEEE Access 8(2020) : 162343 − 162349" and show that the radio mean number of cycles C n determined in Theorem 1 is incorrect when n ≥ 16 by defining radio mean labeling with lower span. We thus derive an improved upper bound on the radio mean number of C n , n ≥ 16. INDEX TERMS Cycle, Radio mean labeling, Radio mean number. I. INTRODUCTION F OR a connected graph G of order n and diameter d(G), a radio mean labeling [2] γ is an assignment of distinct positive integers to the vertices of G satisfying the condition thatwhere d(p, q) is the distance between p and q in G, d(G) is the diameter of G and γ(p), γ(q) are the labels or integers assigned to vertices p and q, respectively. The span of a radio mean labeling γ, denoted by span(γ) is defined as max p∈V (G) γ(p). The radio mean number of G, denoted by rmn(G) is min γ span(γ) where the minimum is taken over all radio mean labelings of G. If rmn(G) = n, then G is said to have a graceful radio mean labeling. E.Badr et al. determined radio mean number of paths and cycles in [1]. However,in [3], authors have shown the existence of radio mean labeling of path graphs with a span less than the one discussed in [1], thereby disproving Theorem 2 of [1]. In this article, we discuss the incorrect radio mean number of cycles determined in Theorem 1 of [1]. We prove that it is erroneous and provide a radio mean labeling with lesser span and thus obtain an improved upper bound on the radio mean number of cycles of order greater than or equal to 16.
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