3-PRODUCT Cordial Labeling of Some Graphs

Proyecciones (Antofagasta)

A mapping f : V (G) → {0, 1, 2} is called 3-product cordial labeling if |v f (i) − v f (j)| ≤ 1 and |e f (i) − e f (j)| ≤ 1 for any i, j ∈ {0, 1, 2}, where v f (i) denotes the number of vertices labeled with i, e f (i) denotes the number of edges xy with f (x)f (y) ≡ i(mod 3). A graph with 3product cordial labeing is called 3-product cordial graph. In this paper we establish that switching of an apex vertex in closed helm, double fan, book graph K 1,n × K 2 and permutation graph P (K 2 + mK 1 , I) are 3-product cordial graphs.

2016

For a graph, a function is called an edge product cordial labeling of G, if the induced vertex labeling function is defined by the product of the labels of the incident edges as such that the number of edges with label 1 and the number of edges with label 0 differ by at most 1 and the number of vertices with label 1 and the number of vertices with label 0 differ by at most 1. In this paper, we show that the graphs obtained by duplication of a vertex, duplication of a vertex by an edge or duplication of an edge by a vertex in a crown graph are edge product cordial. Moreover, we show that the graph obtained by duplication of each of the vertices of degree three by an edge in a gear graph is edge product cordial. We also show that the graph obtained by duplication of each of the pendent vertices by a new vertex in a helm graph is edge product cordial.

Open Journal of Discrete Mathematics, 2025

In 2012, Ponraj et al . defined a concept of k-product cordial labeling as follows: Let f be a map from V (G) to {0,1,, k −1} where k is an integer, 1 ≤ k ≤ V (G) . For each edge uv assign the label f (u) f (v)(mod k ) . f is called a k-product cordial labeling if ( ) ( ) 1 f f v i − v j ≤ , and ( ) ( ) 1 f f e i − e j ≤ , i, j ∈{0,1,, k −1} , where ( ) f v x and ( ) f e x denote the number of vertices and edges respectively labeled with x ( x = 0,1,, k −1). Motivated by this concept, we further studied and established that several families of graphs admit k-product cordial labeling. In this paper, we show that the path graphs n P admit k-product cordial labeling.

Applied Mathematics and Nonlinear Sciences, 2017

In this paper, we study 3–total edge product cordial (3–TEPC) labeling which is a variant of edge product cordial labeling. We discuss Web, Helm, Ladder and Gear graphs in this context of 3–TEPC labeling. We also discuss 3–TEPC labeling of some particular examples with corona graph.

2018

A mapping $f: V(G)rightarrowleft{0, 1, 2 right}$ is called 3-product cordial labeling if $vert v_f(i)-v_f(j)vert leq 1$ and $vert e_f(i)-e_f(j)vert leq 1$ for any $ i, jin {0, 1, 2}$, where $v_f(i)$ denotes the number of vertices labeled with $i, e_f (i)$ denotes the number of edges $xy$ with $f(x)f(y)equiv i(mod 3)$. A graph with 3-product cordial labeling is called 3-product cordial graph. In this paper we establish that vertex switching of wheel,gear graph and degree splitting of bistar are 3-product cordial graphs.

Southeast Asian Bulletin of Mathematics,, 2015

A mapping : () → {0, 1, 2} is called 3-product cordial labeling if | ()− ()| ≤ 1 and | ()− ()| ≤ 1 for any , ∈ {0, 1, 2}, where () denotes the number of vertices labeled with , () denotes the number of edges with () () ≡ (3). A graph with 3-product cordial labeling is called 3-product cordial graph. In this paper, we establish that the splitting graphs ′ (1,), ′ (,), * , and the shadow graph 2(,) are 3-product cordial graphs.

Journal of Applied Mathematics and Computational Mechanics, 2019

For a graph G = (V (G), E(G)) having no isolated vertex, a function f : E(G) → {0, 1} is called an edge product cordial labeling of graph G, if the induced vertex labeling function defined by the product of labels of incident edges to each vertex be such that the number of edges with label 0 and the number of edges with label 1 differ by at the most 1 and the number of vertices with label 0 and the number of vertices with label 1 also differ by at the most 1. In this paper we discuss the edge product cordial labeling of the graphs W (t) n , PS n and DPS n .

Let G be a (p,q) graph and A be a group. Let f : V (G) → A be a function. The order of u ∈ A is the least positive integer n such that un = e. We denote the order of u by o(u). For each edge uv assign the label 1 if (o(u), o(v)) = 1 or 0 otherwise. f is called a group A Cordial labeling if |vf (a) − vf (b)| ≤ 1 and |ef (0) − ef (1)| ≤ 1, where vf (x) and ef (n) respectively denote the number of vertices labeled with an element x and number of edges labeled with n(n = 0, 1). A graph which admits a group A Cordial labeling is called a group A Cordial graph. In this paper we define group {1,−1, i,−i} Cordial graphs and prove that Hypercube Qn = Qn−1 × K2, Book Bn = Sn × K2, n-sided prism Prn = Cn × K2 and Pn × K3 are all group {1,−1, i,−i} Cordial for all n.

Let G be a (p, q) graph. Let f : V (G) → {1, 2,. .. , k} be a map. For each edge uv, assign the label gcd (f (u), f (v)). f is called k-prime cordial labeling of G if |v f (i) − v f (j)| ≤ 1, i, j ∈ {1, 2,. .. , k} and |e f (0) − e f (1)| ≤ 1 where v f (x) denotes the number of vertices labeled with x, e f (1) and e f (0) respectively denote the number of edges labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is called a k-prime cordial graph. In this paper we investigate 3prime cordial labeling behavior of union of a 3-prime cordial graph and a path P n .

Proyecciones (Antofagasta), 2019

Let G be a (p,q) graph. A mapping f : V (G) → {0, 1, 2} is called 3-product cordial labeling if |vf (i) − vf (j)| ≤ 1 and |ef (i) − ef (j)| ≤ 1 for any i, j ∈ {0, 1, 2},where vf (i) denotes the number of vertices labeled with i, ef (i) denotes the number of edges xy with f(x)f(y) ≡ i(mod3). A graph with 3-product cordial labeling is called 3-product cordial graph. In this paper we investigate the 3-product cordial behavior of alternate triangular snake, double alternate triangular snake and triangular snake graphs.