3-product cordial labeling

<p>The concepts of cordial labeling, signed product cordiality, and logical cordiality have been introduced independently by different researchers as distinct labeling schemes. In this paper, we demonstrate the equivalence of these concepts. Specifically, we prove that a graph $ G $ is cordial if and only if it is signed product cordial, if and only if it is logically cordial. Additionally, we establish that a graph $ G $ admits permuted cordial labeling if and only if it exhibits cubic roots cordial labeling. Furthermore, we leverage this newfound equivalence to analyze the cordiality properties of several standard graphs.</p>

Let G be a (p, q) graph. Let k be an integer with 2 ≤ k ≤ p and f from V (G) to the set {1, 2, . . . , k} be a map. For each edge uv, assign the label |f(u) − f(v)|. The function f is called a k-difference cordial labeling of G if |νf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2 . . . , k}), ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate the 3-difference cordial labeling of wheel, helms, flower graph, sunflower graph, lotus inside a circle, closed helm, and double wheel.

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(mod 3). A graph with 3-product 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 K1,n × K2 and permutation graph P (K2 + mK1, I) are 3-product cordial graphs.

In this paper, we introduce a new labeling called Totally Vertex-Magic Cordial(TVMC) labeling. A graph G(p,q) is said to be TVMC with a constant C if there is a mapping f:V(G)∪E(G)→{0,1} such that [ f(a)+∑b∈N(a)f(ab) ]≡C (mod 2) for all vertices a∈V(G) and |nf(0)−nf(1)| ≤1, where N(a) is the set of vertices adjacent to the vertex a and nf(i)(i=0,1) is the sum of the number of vertices and edges with label i.

On 3-total edge product cordial labeling of a carbon nanotube network

Let f be a function from V (G) to {0,1,2}. Give a numerical value (label) from {0, 1, 2} for every edge uv. f is called a mean cordial labeling if |v_f (i)-v_f (j)|≤1 and |e_f (i)-e_f (j)|≤1, ∀ i,j ∈ {0,1,2} where v_f (x) and e_f (x) represent the number of vertices and edges respectively labeled as x(x = 0,1,2). A graph with mean cordial labeling is called a mean cordial graph. In this paper, we study the mean cordial labeling pattern of shadow-graphs of path graphs D_2 (P_n ), for n≥2

Let G = (V,E) be a graph with p vertices and q edges. A Extended Mean Cordial Labeling of a Graph G with vertex set V is a bijection from V to {0, 1,2} such that each edge uv is assigned the label where ⌈ x ⌉ is the least integer greater than or equal to x with the condition that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges labeled with 1 differ by almost 1. The graph that admits an Extended Mean Cordial Labeling is called Extended Mean Cordial Graph. In this paper, we proved that tree related graphs Hdn, K 1,n, Tgn, <K1,n:n> are Extended Mean Cordial Graphs.

In 2012, Ponraj et al. defined k-product cordial labeling as follows: Let [Formula: see text] be a map from [Formula: see text] to [Formula: see text] where [Formula: see text] is an integer, [Formula: see text]. For each edge [Formula: see text] assign the label [Formula: see text] [Formula: see text]. [Formula: see text] is called a k-product cordial labeling if [Formula: see text], and [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] denote the number of vertices and edges, respectively, labeled with [Formula: see text] [Formula: see text]. A graph that admits k-product cordial labeling is called k-product cordial graph. Later, we proved that several families of graphs are k-product cordial graphs. In this paper, we show that the product of graphs admit k-product cordial labeling.

Given a bijection \(f:V(G) \rightarrow \{1,2,\cdots ,|V(G)|\}\), we associate two integers \(S=f(u)+f(v)\) and \(D=|f(u)-f(v)|\) with every edge uv in E(G). The labeling f induces an edge labeling \(f':E(G) \rightarrow \{0,1\}\) such that for any edge uv in E(G), \(f'(uv)=1\) if \(gcd(S,D)=1\), and \(f'(uv)=0\) otherwise. Let \(e_{f'}(i)\) be the number of edges labeled with \(i \in \{0,1\}\). We say f is SD-prime cordial labeling if \(|e_{f'}(0)-e_{f'}(1)| \le 1\). Moreover G is SD-prime cordial if it admits SD-prime cordial labeling. In this paper, we investigate some new construction of SD-prime cordial graph.

The process of assigning the binary numbers 0 and 1 to the edges of a graph \(G = (V, E)\) through evaluating functions defined on the vertex set V and the edge set E of G using the concepts of prime and Narayana numbers by satisfying cordiality on the edges. This process on the graph G is known as Narayana prime cordial labeling of G and this graph G is called Narayana prime cordial graph. In this article, we compute the Narayana prime cordial labeling of Web graphs and Flower graphs.

In this paper, we investigate some new families of edge sum divisor cordial graphs, face edge sum divisor cordial graphs and total face edge sum divisor cordial graphs.Keywords -Edge sum divisor cordial labelling, Edge sum divisor cordial graph, face edge sum divisor cordial graph, total face edge sum divisor cordial graph, switching of any vertex.

An injective function f : V (G) → {T0, T1, T2, . . . , Tn}, where n = |V (G)| − 1, is said to be a Tribonacci cordial labeling if the induced function f ∗ E(G) →{0, 1} defined by f∗(uv) = (f(u) + f(v)) mod 2 satisfies the condition |ef (0) − ef (1)| ≤ 1 where ef (0) is the number of edges with label 0 and ef (1) is the number of edges with label 1. A graph that admits a tribonacci cordial labeling is called a Tribonacci cordial graph. In this paper, we determined the Tribonacci Cordial Labeling of Triangular Snake Graph T Sn, Double Triangular Snake Graph D(T Sn), Quadrilateral Snake Graph QSn, Double Quadrilateral Snake Graph D(QSn), and Cycle Quadrilateral Snake Graph C(QSn).

A bijection f: V(G) → {1,2,··· ,|V(G)|} induces an edge labeling f∗: E(G) → {0,1} such that for any edge uv in G, f∗(uv) = 1 if gcd(S,D) = 1 and f∗(uv) = 0 otherwise, where S = f(u)+ f(v) and D = |f(u)− f(v)|. The labeling f is called SD-prime cordial labeling if |ef∗(0) − ef∗(1)|≤1. We say that G is SD-prime cordial graph if it admits SD-prime cordial labeling. In this paper, we prove that certain classes of zero-divisor graphs of commutative rings are SD-prime cordial graphs.

A power cordial labeling of a graph G = V G , E G is a bijection f : V G ⟶ 1,2 , … , V G such that an edge e = u v is assigned the label 1 if f u = f v n or f v = f u n , for some n ∈ N ∪ 0 and the label 0 otherwise, and satisfy the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. The graph that admits power cordial labeling is called a power cordial graph. In this paper, we derive some characterizations of power cordial graphs as well as explore NP-complete problems for power cordial labeling. This work also rules out any possibility of forbidden subgraph characterization for power cordial labeling.

In this paper we prove that the split graphs of K1,n and Bn,n are prime cordial graphs. We also show that the square graph of Bn,n is a prime cordial graph while middle graph of Pn is a prime cordial graph for n≥4 . Further we prove that the wheel graph Wn admits prime cordial labeling for n≥8.