Cordial Labeling of Some Pan Graphs Sarang Sadawartea and Sweta Srivastava a Department of Mathematics, Sharda University, Greater Noida, India Abstract We introduce cordial labeling of path joining two copies m-pan graph, when (m = 3, 4). We also investigated cordial labeling of path union of r-copies of 3-pan graph is cordial only for r ≡ 0, 1,2(mod4). Keywords 1 Pan graph, path union of 3-pan graph, cordial graph 1. Introduction Graph theory plays an important role in the field of mathematics, computer science, operation research, physics, chemistry, biology in general and widely in communication. In the field of graph theory labeling has multiple application like networking, data base management, circuit designing, coding, communication networking. Graph labeling is a function that carries set of vertices or edges to non-negative integers. Throughout the paper K is finite, simple and undirected. We follow Harary [5] for graph theory related basic terminology and notations. The idea of cordial labeling was first invented by Cahit [1] 1987. 2. Terminology and Notation 2.1. Definition Let K = (V, E) be a graph and f : V (K) → {0, 1} is said to be a (0−1) or a binary vertex labeling of K and the vertex label of K under f is represented by f(l). Consider the induced mapping f*(kl) : E(K) → {0, 1} and derived as f*(kl) = |f(k) − f(l)| where e = kl. We represent, lf (h) as for any vertex v ∈ V (K) such as f(v) = h and ef (h) as for any edge e ∈ E(K) such as f*(e) = h, where h ∈ {0, 1}. A labeling (0 − 1) of a graph K is a cordial labeling if following criteria satisfies a) |lf (0) − lf (1)| ≤ 1 b) |ef (0) − ef (1)| ≤ 1. A graph K which preserves cordiality known as a cordial graph. [6] 2.2 Definition A resultant graph K is called m-pan graph if it is constructed with joining a cycle graph C m to a pendent vertex. WCNC-2021: Workshop on Computer Networks & Communications, May 01, 2021, Chennai, India. EMAIL: 2020489011.sadawarte@dr.sharda.ac.in (Sarang Sadawarte) ORCID: 0000-0002-4223-6949 (Sarang Sadawarte) © 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) 87 2.3 Definition A path union of K is constructed with attaching r copies of 3-pan graph with the path of size r − 1. 3. Results Theorem 3.1. A graph constructed with attaching 2 copies of 3-pan graph by a path of distinct size preserves cordial labeling. Proof. We shall denote K as a resultant graph constructed with attaching 2 copies of 3-pan graph by a path of distinct size. Let distinct vertices of 1st and 2nd copies of 3-pan graph be denoted by l1, l2, and l3 and k1, k2 and k3 respectively. Let vertices 1st, 2nd and rth of path Pr be represented by x1, x2, . . . , xr which carries condition as first vertex x1 coincides with l1 and rth vertex, xr coincides with k1. Define a vertex labeling mapping f : V (K) → {0, 1} as given below Case I for r ≡ 1(mod4) 0 ; h = 1(mod4) f(lh) = 1 ; h = 2, 3(mod4) 0 ; h = 1(mod4) f(kh) = 1 ; h = 2, 3 (mod4) 0 ; h = 0, 1(mod4) f(xh) = 1 ; h = 2, 3(mod4) Case II For r ≡ 2(mod4) 0 ; h = 1(mod4) f(lh) = 1 ; ℎ = 2, 3(𝑚𝑜𝑑4) 0 ; ℎ = 1(𝑚𝑜𝑑4) f(kh) = 1 ; h = 2, 3(mod4) 0 ; h = 1, 2(mod4) f(xh) = 1 ; ℎ = 0, 3(𝑚𝑜𝑑4) Table 1: Edges Labeling Pattern Above table elaborate labeling pattern of edges. Clearly the condition of cordiality |ef (0) − ef (1)| ≤ 1 is satisfied. Hence, the resultant graph K is cordial. 88 Example 3.1. The cordial labeling resultant graph K under certain condition r ≡ 1(mod4) is constructed with Figure 1. Figure 1: Path joining of 2 copies of 3-pan graph Theorem 3.2. A graph constructed with attaching 2 copies of 4-pan graph by a path of distinct size preserves cordial labeling. Proof. We shall denote K as a resultant graph constructed with attaching 2- copies of 4-pan graph by a path of distinct size. Let distinct vertices of 1st and 2nd copies of 4-pan graph be denoted by l1, l2, l3 and l4 and k1, k2, k3 and k4 respectively. Let the vertices 1st, 2nd and rth of path Pr be represented by x1, x2, . . . , xr which carries condition as first vertex x1 coincides with l1 and rth vertex, xr coincides with k1. Define a vertex labeling mapping f : V (K) → {0, 1} as given below: For r ≡ 0(mod4) 0 ; h = 1, 2(mod4) f(lh) = 1 ; h = 0, 3(mod4) 0 ; h = 1, 2(mod4) f(kh) = 1 ; h = 0, 3(mod4) 0 ; h = 0, 1(mod4) f(xh) = 1 ; h = 2, 3(mod4) Table 2: Edges Labeling Pattern Above table elaborate labeling pattern of edges. Clearly the condition of cordiality |ef (0) − ef (1)| ≤ 1 is satisfied. Hence, the resultant graph K is cordial. Example 3.2. The cordial labeling of the resultant graph K under certain condition r ≡ 0(mod4) is elaborated by Figure 2. Figure 2: Path joining of 2 copies of 4-pan graph 89 Theorem 3.3. Any path-union of r-copies of 3-pan graph admits cordiality. Proof. Consider K as a resultant graph constructed with attaching r copies of 3-pan graph by a path of distinct r − 1 size. Let the vertices of 1st, 2nd , . . . , rth copy graph be lh1, lh2, . . . , lhr. Let x1, x2, . . . , xr be the vertices of path Pr with condition x1 = lh1, x2 = lh2, . . . , xr = lhr. Define a vertex labeling mapping f : V (K) → {0, 1} is stated below Case I: For r ≡ 0(mod4) 0 ; h = 1(mod4) f(lhq) = 1 ; ℎ = 2, 3(𝑚𝑜𝑑4) 0 ; h = 2, 3(mod4) f(lhq−2) = 1 ; h = 1(mod4) Case II: For r ≡ 1(mod4) 0 ; h = 1(mod4) f(lhq) = 1 ; ℎ = 2, 3(𝑚𝑜𝑑4) 0 ; h = 2, 3(mod4) f(lhq−2) = f(lhq−3) = 1 ; h = 1(mod4) Case III: For r ≡ 2(mod4) 0 ; ℎ = 1(𝑚𝑜𝑑4) f(lhq) = 1 ; ℎ = 2, 3(𝑚𝑜𝑑4) 0 ; h = 2, 3(mod4) f(lhq−2) = f(lhq−3) = 1 ; ℎ = 1(𝑚𝑜𝑑4) Table 3: Edges and Vertices Labeling Above table elaborate labeling pattern of edges. Clearly the condition of cordiality |ef (0) − ef (1)| ≤ 1 is satisfied. Hence, any path union of r-copies of 3-pan graph admits cordial labeling. Example 3.3 Cordial labeling of path-union of r-copies of 3-pan graph under certain condition r ≡ 2(mod4) is shown in Figure 3. Figure 3: Path union of 6-copies of 3-pan graph 90 4. Conclusion In this paper we emphasized path joining two copies m-pan graph and path-union of r copies of 3- pan graph is cordial. To elaborate several families of graph which satisfies corresponding results is an open problem for the further research. In the field of graph theory labeling has multiple application like networking, data base management, circuit designing, coding, communication networking. Graph theory plays an important role in the field of mathematics, computer science, operation research, physics, chemistry, biology in general and widely in communication. 5. Acknowledgement Author and co Author would like to thank Cahit. I. for introducing the concept of cordial labeling, using that we have generated cordial labeling of some pan graphs. 6. 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