The computational complexity of cordial and equitable labelling

Graph labeling is the term used to describe the procedure of giving unique identifiers. Graph attributing is a method for modeling real-world problems in industries such as circuit design, surveillance technology, and telecommunications networks. A graph is characterized as possessing cubic roots cordial labeling if each node is designated with either the value 1, δ or δ2 or another specified integer. The derived set containing the value 1 or another integer, the mathematical conditions η (k) − η (w)| ≤ 1 and | ξ (k) − ξ (w)| ≤ 1, k ≠ w and k, w ∈ {1, δ, δ2} where η (k) and ξ (k) are represents the aggregated vertices and edges assigned in k ∈ {1, δ, δ2}. Labeled graphs prove to be immensely beneficial across a variety of mathematical frameworks, including domains such as cyber security and radar code formulation. The application of cordial labeling has been demonstrated to provide significant advantages in the context of signal processing channels and challenges associated with specific code word design. The idea of cubic roots cordial labeling has attracted a lot of academic attention among various labeling techniques, unique structural characteristics and capacity to depict a fair flow of information within linked networks. The investigation of the behavior of cubic roots cordial labeling in relation to graph operations is currently receiving extensive scholarly examination, particularly concerning path graphs, despite its critical relevance. This paper explore the cordial labeling of cubic product graphs generated from the combination of two path graphs. The four fundamental graph operations on which we particularly focus on activities, we analyze the specific conditions to possess cubic roots cordial labeling.

Assigning an integer to a vertices or edges is called a vertex or edge labeling respectively. Suppose G is a simple graph. Consider the function f for the edge set . For each vertex t , define f(t)= (mod2). The function f is known as an E-cordial labeling (E-CL) of G if , and where , and , are the number of edges and vertices labeled with 0 and labeled by 1 respectively. A graph that admits E-CL is said to be E-cordial graphs (E-CG). Based on the above definition we propose a novel labeling known as SP Mean E-cordial labeling (E-CL). In this paper, we study SP Mean E-CL of several families of graphs such as complete bipartite graphs, complete graphs, wheels, etc.

Let G be a simple graph with vertex set V(G) and edge set E(G). Let ⟨ℤ2, +,*⟩ be a field with two elements. A vertex labeling f : V(G) → ℤ2 induces two edge labelings f+: E(G) → ℤ2 such that f+ (xy) = f(x) + f(y), whereas f* : E(G) → ℤ2 such that f* (xy) = f(x) f(y), for each edge xy ∈ E(G). For i ϵ ℤ2, let and . A labeling f of a graph G is said to be friendly if |υf (0) −υf (1)|≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as the vertex labeling f is friendly}. This is a generalization of graph cordiality. The corresponding multiplicative version is called the product-cordial index set, denoted PCI(G), defined as the vertex labeling f is friendly}. In this paper, we investigate the friendly index and product-cordial index sets of a family of cubic graphs known as Möbius-liked graph, MG(n) for even n ≥ 4.

Cordial labeling of hypertrees

An $L(2,1)$-labeling of a graph is a mapping $c:V(G) \to \{0,\ldots,K\}$ such that the labels assigned to neighboring vertices differ by at least $2$ and the labels of vertices at distance two are different. Griggs and Yeh [SIAM J. Discrete Math. 5 (1992), 586―595] conjectured that every graph $G$ with maximum degree $\Delta$ has an $L(2,1)$-labeling with $K \leq \Delta^2$. We verify the conjecture for planar graphs with maximum degree $\Delta \neq 3$.

The product cordial labeling is a variant of cordial labeling. Here we investigate product cordial labelings for alternate triangular snake and alternate quadrilateral snake graphs.

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.

An $$L(j,k)$$L(j,k)-labeling of a graph is a vertex labeling such that the difference of the labels of any two adjacent vertices is at least $$j$$j and that of any two vertices of distance $$2$$2 is at least $$k$$k. The minimum span of all $$L(j,k)$$L(j,k)-labelings of $$G$$G is denoted by $$\lambda _k^j(G)$$?kj(G). Lin and Lam (Discret Math 308:3805---3815, 2008) provided an upper bound of $$\lambda _1^2(K_m \times C_n)$$?12(Km×Cn) when $$K_m \times C_n$$Km×Cn is the direct product of a complete graph $$K_m$$Km and a cycle $$C_n$$Cn. And they found the exact value of $$\lambda _1^2(K_m \times C_n)$$?12(Km×Cn) for some $$m$$m and $$n$$n. In this paper, we obtain an upper bound and a lower bound of $$\lambda _1^1(K_m \times C_n)$$?11(Km×Cn). As a consequence we compute $$\lambda _1^1(K_m \times C_n)$$?11(Km×Cn) when $$n$$n is even or $$n\ge 4m+1$$n?4m+1.

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 graph theory many useful mathematical disciplines, but the extremely useful one is graph labeling. A graph labeling is used in many applications in various fields like X-ray crystallography, computer science, data base management, coding theory, astronomy and physics. A labeling of edges and vertices of a simple graph G(V, E) by a mapping ¥ : V(G) ∪ E(G) → {1, 2, 3, … , ℜ} provided that any two different edges have distinct weights is called an edge irregular total ℜ -labeling. If ℜ is minimum and G admits an edge irregular total ℜ-labeling, then it is called the total edge irregularity strength (TEIS) denoted by tes(G). In this paper, we have introduced three definitions: the quintet snake graph QS n , the double quintet snake graph D(QS n) and the m–multiple quintet snake graph M(QS n). Moreover, the exact value of TEIS for the new graphs are investigated.

<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>

A (0; 1)-labeling of a set is said to be friendly if the number of elements of the set labeled 0 and the number labeled 1 differ by at most 1. Let g be a labeling of the edge set of a graph that is induced by a labeling f of the vertex set. If both g and f are friendly then f is said to be a cordial labeling of the graph. This concept extended to directed graphs is called (2; 3)-cordiality of digraphs. We investigate the labelings that are both cordial for a graph and (2; 3)-cordial for an orientation of it. We also consider the same problem for other known binary vertex labelings of graphs.

This study investigates the application of Smarandache product cordial labeling techniques to Christmas tree graphs. Graph labeling is a crucial aspect of graph theory, with applications in various fields such as computer science, coding theory, and network design. Smarandache product cordial labeling, in particular, offers a unique approach to labeling graphs while maintaining specific properties. By exploring the application of these techniques to Christmas tree graphs, this research aims to contribute to the understanding of graph labeling and its potential applications in real-world problems. The study presents theoretical results, algorithmic approaches, and potential extensions to related graph structures.

This papers deals with the Edge cordial labeling of newly introduced eight sprocket graph. This graph is already proven as cordial and gracious in graph labelling. In our study we have further proved that Eight Sprocket graph related families of connected edge-cordial graphs. Also the path union of Eight Sprocket graph, cycle of Eight Sprocket graph and star of Eight Sprocket graph are Edge-cordial.Â

Many program-analysis problems can be formulated as graph-reachability problems. Interleaved Dyck language reachability ( InterDyck -reachability) is a fundamental framework to express a wide variety of program-analysis problems over edge-labeled graphs. The InterDyck language represents an intersection of multiple matched-parenthesis languages (i.e., Dyck languages). In practice, program analyses typically leverage one Dyck language to achieve context-sensitivity, and other Dyck languages to model data dependencies, such as field-sensitivity and pointer references/dereferences. In the ideal case, an InterDyck -reachability framework should model multiple Dyck languages simultaneously . Unfortunately, precise InterDyck -reachability is undecidable. Any practical solution must over-approximate the exact answer. In the literature, a lot of work has been proposed to over-approximate the InterDyck -reachability formulation. This article offers a new perspective on improving both the precision and the scalability of InterDyck -reachability: we aim at simplifying the underlying input graph G . Our key insight is based on the observation that if an edge is not contributing to any InterDyck -paths, we can safely eliminate it from G . Our technique is orthogonal to the InterDyck -reachability formulation and can serve as a pre-processing step with any over-approximating approach for InterDyck -reachability. We have applied our graph simplification algorithm to pre-processing the graphs from a recent InterDyck -reachability-based taint analysis for Android. Our evaluation of three popular InterDyck -reachability algorithms yields promising results. In particular, our graph-simplification method improves both the scalability and precision of all three InterDyck -reachability algorithms, sometimes dramatically.