Research article · 2017 · Full text
International Journal of Statistics and Applied MathematicsNew families and invariance under E-cordial labeling
Volume 2(6) · Pages 151–154
Research areasGraph Labeling and Dimension ProblemsFuzzy and Soft Set TheoryAdvanced Mathematical Theories
Abstract
Author-writtenIn this paper we discuss new families of E- cordial labeling. These are FL(Sn), (S4)(k), Crown of shel, (Sn)(k), (K1,n:K1,n), (FL(C4) )(k), E-cordial invariance of (FL(C3) )(k)
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- E-cordial labeling was introduced in 1997 as a weaker version of edge-graceful labeling. View evidence ↗
- A graph is considered E-cordial if there exists a labeling where the counts of vertices labeled 0 and 1 differ by at most 1, and the counts of edges labeled 0 and 1 also differ by at most 1. View evidence ↗
The question
- The study aims to investigate E-cordial labeling for several new graph families, including flag graphs of shells and one-point unions of specific graph copies. View evidence ↗
What they did
- The research focuses on graphs that are simple, connected, undirected, and finite. View evidence ↗
- The vertex labels are derived from the sum of the labels of the incident edges modulo 2. View evidence ↗
- Flag graphs (FL(G)) are constructed by attaching a copy of K2 to a specific vertex of the base graph G. View evidence ↗
What they found
- The flag graph of a shell, FL(Sn), is E-cordial if and only if n is an even number. View evidence ↗
- The one-point union of k copies of S4 is generally E-cordial, except when the number of vertices is congruent to 2 (mod 4). View evidence ↗
- The graph (K1,n: K1,n) is E-cordial for both even and odd n, with edge distributions always being equal (ef(0)=ef(1)=n). View evidence ↗
- The one-point union of k copies of FL(C3) is E-cordial across three different structural configurations, except when 3k is congruent to 1 (mod 4). View evidence ↗
- Because the E-cordiality of (FL(C3))(k) holds regardless of which of the three possible union points is used, the graph is considered invariant under E-cordial labeling. View evidence ↗
- The one-point union of k copies of FL(C4) is E-cordial, with vertex distributions of 2k+1 and 2k. View evidence ↗
What it means
- The authors conclude that (FL(C3))(k) is invariant under E-cordial labeling. View evidence ↗
- The authors suggest that future research should investigate whether the same invariance applies to (FL(Cn))(k) for any given n. View evidence ↗
Visual summary
2 visualsIllustrations explain the paper; charts are replotted from the paper's own table values.
E-Cordial Labeling Concept
AI-generated conceptual illustration (mathematical graph consisting of vertices and edges). It explains the idea and does not show study data.
View source evidence ↗
vf(0) and vf(1) and more: 1 vs 2 vs 3* vs 4
Replotted from Table 1 (page 2) of the original paper. Every bar is the paper's own table value.
View source evidence ↗Figures & tables
From the original paper
Table 1
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Table 2
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Figure 8Introduction
In 1997 Yilmaz and Cahit [5] introduced a weaker version of edge-graceful called E-cordial. labeling. Let G be a graph with vertex set V and edge set E and let f a function from E to {0, 1}. Define f on V by f(v) =Σf(uv) (mod 2). The function f is called an E-cordial labeling of G if the number of vertices labeled 0 and the number of vertices labeled 1 differ by at most 1 and the number of edges labeled 0 and the number of edges labeled 1 differ by at most 1. A graph that admits an E-cordial labeling is called E-cordial. The graphs considered are finite, undirected, simple and connected. For terminology and definations we refer Harary [4] , Dynamic survey of graph labelling [3] . Further by vf(0, 1)=(a, b) we mean number of vertices with label 0 are a in number and that with label 1 are b in number. Similar convention is followed for edge numbers ef(0, 1) = (x, y).
Definitions 3.1.
Shel Sn is obtained from Cn by taking n-3 chords starting with the same vertex on Sn say v and ending on the vertices of Cn that are not adjacent to v. 3.2. FL(G) is a flag graph of G and is obtained by attaching a copy of k2to a suitable vertex of G. If This vertex is changed then the graph may differ structurally. For FL(Cn) and FL(Kn) which vertex is used to attach K2 does not matters. For Flag of Sn we attach K2 at apex of Sn.
3.3.
Antena graph Consider a G = (p, q) graph. At each of it's vertex attach a path of length m. then we get a antenna graph antenna (G, m). If we attach K antennas of different length at each vertex of G then it is k-antena(G).
Results proved 4.1. Theorem: Fl(Sn) is E-cordial iff n is even number.
Proof: Ordinary names given to vertices are that u1 is the apex vertex and main cycle Cn of Sn is given by (u1, e1, u2, e2,.. en, u1). The chords are given by ci=(u1uj),i = j-2 and j = 3, 4,.., n-2. Define a function f;E→{0,1}by f(ei)=0 for i is a even number and f(ei)= 1 for i>1and is an odd number. f(e1)=0. The chords n-3 in number are labeled as f(ci) =1 for i is an odd number and f(ei) = 0 for i is an even number. The final vertex numbers distribution we get is vf(0, 1)=(x+1, x) for p≡1(mod 4) and vf(0, 1)= (x, x+1) for p≡3(mod 4). and edge numbers are ef(0, 1) = (n-1, n-1). The table 1 explains the construction of G from using different types (A, B, C, D) of labels and resultant vertex and edge distribution. Note that * in the first column indicate that for that value of k the graph is not E-cordial. The note made by Yilmaz and Cahit is also observed here, that for number of vertices congruent to 2(mod 4), the graph is not E-cordial. Label all the pendent vertices incident with u as 1 and that at v as 0.edge (uw) is labeled as 0 and edge (wv) as 1.
| k | Type Used | vf(0) | vf(1) | ef(0) | ef(1) | Comm-on vertex. |
|---|---|---|---|---|---|---|
| 1 | A | 2 | 2 | 2 | 3 | 0 |
| 2 | A+C | 3 | 4 | 5 | 5 | 0 |
| 3* | A+C+B | 4 | 6 | 8 | 7 | 1 |
| 4 | A+C+B+D | 7 | 6 | 10 | 10 | 0 |
| 4X | x times A, C, B, D | 6X+1 | 6X | 10X | 10X | 0 |
| 4X+1 | x times A, C, B, D +A | 6X+2 | 6X+2 | 10X+2 | 10X+3 | 0 |
| 4X+2 | x times A, C, B, D +A+C | 6X+3 | 6X+4 | 10X+5 | 10X+5 | 0 |
| 4X+3* | x times A, C, B, D +A+C+B | 6x+6 | 6x+4 | 10x+8 | 10x+7 | 1 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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The label numbers are given by: for n is even vf(0)=n+1 vf(1)= n on vertices. ef(0)=n = ef(1) on edges.
For n is odd, vf(0) = n, vf(1)= n+1 on vertices. ef(0)=n = ef(1) on edges. Thus G is E-cordial.
Theorem:
All three structures of G = (FL (C3)) (k) i.e. One point union of k-copies of FL( C3) is E-cordial except for 3k≡1(mod 4) Proof: There can be three distinct points on FL(C3)which are used in three cases to obtain one point union. These points on FL(C3) are i) pendent vertex ii) degree 2 vertex iii) degree 3 vertex. We discuss three cases separately. Case 1: The common point is pendent vertex u.
The union point is taken on pendent vertex of FL(C3).The i th copy of FL(C3) is defined as (u ei1 ui2 ei2 ui3 ei3 ui4 ei4 ui2) i =1,2, k. We use two type of labeling namely type A and Type B to design G. Fig. 4 gives details. The Edge distribution is always ef(0)=2k=ef(1) for every k.
The vertex distribution we have, (for k>4) For k ≡ 0(mod 4) we have vf(0)= 3k/2+1,vf(1)=vf(0)-1 For k ≡ 1(mod 4) vf(0)=(3k+1)/2; vf(1)=vf(0) For k ≡ 2(mod 4) vf(0)=3k/2; vf(1)=vf(0)+1 For k ≡ 3(mod 4) vf(0)=(3k+3)/2; vf(1)=vf(0)-2 Thus except for k ≡3(mod 4) we have G is Ecordial. This is equivalent to observation if n≡2 (mod 4) the graph is not E-cordial. Where n is number of vertices on graph and is given by 3k+1.
Figures and tables from the paper
| k | Type of label | f(u). It is label of common vertex | vf(0) | vf(1) | ef(0)=ef(1) |
|---|---|---|---|---|---|
| 1 | A | 1 | 2 | 2 | 2k |
| 2 | A+B | 1 | 3 | 4 | 2k |
| 3* | A+B+A | 0 | 6 | 4 | 2k |
| 4 | A+B+A+B | 1 | 7 | 6 | 2k |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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References
3 in the original paper- Bapat MV. Ph. D. Thesis, University of Mumbai 2004.
- Gallian J. Electronic Journal of Graph Labeling (Dynamic survey). 2016.
- Yilmaz R, Cahit I. E-cordial graphs, Ars Combin. 1997; 46:251-266.