k-Odd Sequential Harmonious Labeling of Some Special Graphs

Southeast Asian Bulletin of Mathematics,, 2019

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, • • • , 2q − 1} such that the induced function f * : E(G) → {1, 3, • • • , 2q − 1} defined by f * (uv) = f (u) + f (v) is a bijection. In this paper we prove that m-shadow and m-splitting of the graphs Pn, Hn,n, Kr,s, Pn ⊕ K2 and Splm(Cn), n ≡ 0(mod 4) are odd harmonious graphs.

2020

A graph G is a 2-odd graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the absolute difference of their labels is either an odd integer or 2. In this paper, we investigate 2-odd labeling of various classes of graphs.

International Journal of Mathematics Trends and Technology, 2017

In this paper some new type of odd graceful graphs are investigated. We prove that the graph obtained by joining a cycle C 8 with some star graphs S 1,r keeping one vertex and three vertices gap between pair of vertices of the cycle admits odd graceful labelling. We also prove that the graph obtained by joining a cycle C 12 with some star graphs S 1,r keeping two, three and five vertices gap between pair of vertices of the cycle admits odd graceful labelling. We observed that C 8 ʘ S 4 1,r with one vertex gap, C 8 ʘ S 2 1,r three vertices gap between pair of vertices of the cycle C 8 are odd graceful , C 12 ʘ S 6 1,r with one vertex gap, C 12 ʘ S 4 1,r two vertices gap, C 12 ʘ S 3 1,r with three vertices gap and C 12 ʘ S 2 1,r with five vertices gap between pair of vertices are odd graceful graphs.

Southeast Asian Bulletin of Mathematics, 2022

A graph G(p, q) is said to be odd harmonious if there exists an injection f : V (G) → {0, 1, 2, • • • , 2q − 1} such that the induced function f * : E(G) → {1, 3, • • • , 2q − 1} defined by f * (uv) = f (u) + f (v) is a bijection. In this paper we prove that path union of r copies of Km,n, path union of r copies of Km i ,n i , 1 ≤ i ≤ r, K t m,n , K t (m 1 ,n 1),(m 2 ,n 2),••• ,(mt,nt) , join sum of graph Km,n; Km,n; • • • , Km,n(t copies) , Km 1 ,n 1 ; Km 2 ,n 2 ; • • • , Km t ,nt , circle formation of r copies of Km,n when r ≡ 0 (mod 4), S(t.Km,n) and P t n (t.n.Kp,q) are odd harmonious graphs.

A graph , is called , -graph if it has vertices and edges. A , -graph is said to be an odd harmonious if there exists an injection : → 0, 2 1 such that the induced * is a bijection from onto 1, 2 1 1,3,5, … ,2 1 and is said to be an odd harmonious labeling of . A graph is called kC -snake graphs with k 1 is a connected graph with k blocks whose block-cutpoint graph is a path and each of the k blocks is isomorphic to C .In this paper, we constructthe odd harmonious labeling on kCn-snake graph for several values of n. Moreover, we also give odd harmonious labeling construction for a class of graph which is variation of kCn-snake graph.

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE , 2017

A graph G(p,q) is said to be odd harmonious if there exists an injection f : V (G) → {0,1,2,··· ,2q − 1} such that the induced function f ∗ : E(G) → {1,3,··· ,2q − 1} defined by f ∗ (uv) = f(u) + f(v) is a bijection. In this paper we prove that the plus graph Pl n , open star of plus graph S(t.Pl n ), path union of plus graph Pl n , joining of C m and plus graph Pl n with a path, one point union of path of plus graph P t n (t.n.Pl n ) are odd harmonious graphs.

2014

In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.

Cubo (Temuco), 2020

A graph G(p, q) is said to be odd harmonious if there exists an injection f : In this paper we prove that T p -tree, m and subdivided grid graphs are odd harmonious. Un grafo G(p, q) se dice impar armonioso si existe una inyección f : es una biyección. En este artículo probamos que los grafos T p -árboles, T •P m , T •2P m , árboles bambú regulares, C n •P m , C n •2P m y cuadrículas subdivididas son impar armoniosos.

Harmonious labeling of graph is getting lots of application in social networking, rare probability event and many more. Here we will discuss about some harmonious labeling techniques and some important theorems and examples based on those theorems.

Mathematics and Statistics, 2022

Graph theory plays a significant role in a variety of real-world systems. Graph concepts such as labeling and coloring are used to depict a variety of processes and relationships in material, social, biological, physical, and information systems. Specifically, graph labeling is used in communication network addressing, fault-tolerant system design, automatic channel allocation, etc. 2-odd labeling assigns distinct integers to the nodes of G(V, E) in such a manner, that the positive difference of adjacent nodes is either 2 or an odd integer, 2k ± 1, k ∈ N. So, G is a 2-odd graph if and only if it permits 2-odd labeling. Studying certain important modifications through various graph operations on a given graph is interesting and challenging. These operations mainly modify the underlying graph's structure, so understanding the complex operations that can be done over a graph or a set of graphs is inevitable. The motivation behind the development of this article is to apply the concept of 2-odd labeling on graphs generated by using various graph operations. Further, certain results on 2-odd labeling are also derived using some well-known number theoretic concepts such as the Twin prime conjecture and Goldbach's conjecture, besides recalling a few interesting applications of graph labeling and graph coloring.