The Harmonious, Odd Harmonious, And Even Harmonious Labeling

Abstract

Suppose 𝐺 is a simple and connected graph with 𝑞 edges. A harmonious labeling on a graph 𝐺 is an injective function 𝑓: 𝑉(𝐺) → {0, 1, 2, … , 𝑞 − 1} so that there exists a bijective function 𝑓 ∗ : 𝐸(𝐺) → {0,1, 2, … , 𝑞 − 1} where 𝑓 ∗(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣)(𝑚𝑜𝑑 𝑞), for each 𝑢𝑣 ∈ 𝐸(𝐺). An odd harmonious labeling on a graph 𝐺 is an injective function 𝑓 from 𝑉(𝐺) to non-negative integer set less than 2𝑞 so that there is a function 𝑓 ∗(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣) where 𝑓 ∗(𝑢𝑣) ∈ {1, 3, 5, … , 2𝑞 − 1} for every 𝑢𝑣 ∈ 𝐸(𝐺). An even harmonious labeling on a graph 𝐺 is an injective function 𝑓: 𝑉(𝐺) → {0, 1, 2, … , 2𝑞} so that there is a bijective function 𝑓 ∗ : 𝐸(𝐺) → {0,2,4, … , 2𝑞 − 2} where 𝑓 ∗(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣)(𝑚𝑜𝑑 2𝑞) for each 𝑢𝑣 ∈ 𝐸(𝐺). In this paper, we discuss how to build new labeling (harmonious, odd harmonious, even harmonious) based on the existing labeling (harmonious, odd harmonious, even harmonious).