Research article Β· 2018 Β· Full text
International Journal of Statistics and Applied MathematicsPath and cycle related families of E-Cordial graph
Volume 3(1) Β· Pages 263β265
Research areasGraph Labeling and Dimension ProblemsGraph theory and applicationsAdvanced Graph Theory Research
Abstract
Author-writtenThe two copies of graph G(p, q) are joined by t paths on n-points each We represent the family by G(tPn). The paths are attached at the same fixed point on G. We discuss E-cordiality of C3(Pn), K4(Pn), S4(Pn) (shel graph S4 ) We show that under certain conditions these graphs are E-cordial.
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 by Yilmaz and Cahit in 1997 as a weaker version of edge graceful labeling. View evidence β
- A labeling is considered E-cordial if the difference between the number of vertices labeled 0 and 1 is at most 1, and the difference between the number of edges labeled 0 and 1 is also at most 1. View evidence β
- It has been observed that any graph with n vertices cannot be E-cordial if n is congruent to 2 modulo 4. View evidence β
The question
- The study aims to determine the conditions under which the graph families C3(Pn), K4(Pn), and S4(Pn) are E-cordial. View evidence β
What they did
- The research focuses on families G(tPn) created by joining two copies of a graph G (specifically C3, S4, and K4) using a single path (t=1) of equal length attached at a fixed point. View evidence β
- The labeling method assigns values from {0, 1} to the edges, and the vertex labels are derived as the sum of the labels of incident edges modulo 2. View evidence β
What they found
- The graph G=C3(1Pn) is E-cordial if and only if the total number of vertices n' is not congruent to 2 modulo 4. View evidence β
- The shel graph family S4(Pn) is E-cordial if and only if n is not congruent to 2 modulo 4. View evidence β
- The graph family K4(Pn) is found to be E-cordial. View evidence β
- For the S4(Pn) family, the graph is not E-cordial if n is divisible by 4. View evidence β
What it means
- The authors conclude that while they analyzed single paths between graph copies, further evaluation is needed for cases with two or more paths. View evidence β
- The scope of the current study was limited to the analysis of a single path connecting the two copies of the graphs. View evidence β
Visual summary
1 visualsIllustrations explain the paper; they do not show study data.
Graph Family Construction
AI-generated conceptual illustration (Abstract mathematical graph networks consisting of nodes and edges). It explains the idea and does not show study data.
View source evidence βIntroduction
In 1997 Yilmaz and Cahit introduced weaker version of edge graceful labeling E-cordial labeling. Let G be a (p, q) graph. f:Eο {0, 1} Define f on V by f(v) =β{π(π£π’)(π£π’)ΠπΈ(πΊ)}(πππ 2). The function f is called as E-cordial labeling if |vf(0)-vf(1)|β€1 and |ef(0)-ef(1)|β€1 where vf(i) is the number of vertices labeled with i =0, 1. And ef(i) is the number of edges labeled with i= 0, 1, We follow the convention that vf(0, 1) =(a, b) for vf(0)=a and vf(1)=b further ef(0,1)=(x, y) for ef(0)=x and ef(1)=y.
A graph that admits E-cordial labeling is called as E-cordial graph. Yilmaz and Cahit prove that trees Tn are E-cordial iff for n not congruent to 2(mod 4), Kn are E-cordial iff n not congruent to 2(mod 4), Fans Fn are E-cordial iff for n not congruent to 1(mod 4). Yilmaz and Cahit observe that A graph on n vertices can not be E-cordial if n is congruent to 2 (mod 4). One should refer Dynamic survey of graph labeling by Joe Gallian [2] for more results on E-cordial graphs.
The graphs we consider are finite, undirected, simple and connected. For terminology and definitions we refer Harary [3] and Dynamic survey of graph labeling by Joe Gallian [2]. The families we discuss are obtained by taking two copies of graph G and join them by t paths of equal length. The paths are attached at the same fixed point on G. we represent these families by G(tPn).
We take t = 1 and G = C3, S4., K4. Theorems proved G=C3(1Pn) is E-cordial iff nβ is not congruent to 2 (mod 4).where nβ=|V(G)| Proof: We define Gas V(G)={u1, u2, u3, u4}U{v1, v2,.vn} E(G)={ei=(vivi+1), i=1, 2, n-1, c1=(u1u2) c2=(u2v1),c3=(u1v1), c4=(vnu3), c5=(u3u4), c6=(u4vn)} |V(G)| =n+4 and |E(G)| = n+3. Maths 2018; 3(1): 263-265 Mukund V Bapat At Post Hindale, Tal Devgad, Dist. Sindhudurg, Maharashtra, India Correspondence Mukund V Bapat At Post Hindale, Tal Devgad, Dist.
Sindhudurg, Maharashtra, India πβπ For n>5 define a function f:V(G)ο {0, 1} as f(u1)=f(u2)=0 f(c4)=f(c5)=f(c6)=0,f(c1)=f(c3)=0, f(c2)=1 for i β€2t; f(ei) =1 for I is odd number f(ei)=0 I is even number Where t =min{ πβπ 4 /i=0,1,2,3} Theorem S4(Pn) is E-cordial iff n is not congruent to 2 (mod 4) Proof: S4(pn)is defined as V(G) = {u1, u2, u3, u4, u5, u6}U{v1, v2, v3.vn} and the edge set is given by E(G) = {c1=(v1u1), (u1u2), c3=(u2u3), c4=(u3u4), c4=(u4v1), c5=(v1u2)} U {ei = (vivi+1), i= 1, 2,β¦,n-1} For n>5 define a function f:V(G)ο {0,1} as f(u1)=f(u2)=0 f(c4)=f(c5)=f(c6)=0, f(c1)=f(c3)=0, f(c2)=1 for i β€2t; f(ei) =1 for I is odd number f(ei)=0 I is even number Where t =min{ 4 /i=0, 1, 2, 3} πβπ Define a function f : V(G)ο {0, 1} as f(u1)=f(u2)=0 f(c4)=f(c5)=f(c6)=0, f(c1)=f(c3)=0,f(c2)=1 take t = πβ1 4 /j=1, 2, 3. (n is divisible by 4 then graph is not E- cordial) For i β€2t f(ei)=0 for i is odd number β€2t f(ei)=1 for i is even number <2t. f(ei)=1 for i= 2t, 2t+1, 2t+k where k = p-t+1 where p = 2 for πβ2 n β‘1,3 (mod 4) and p = π π π π 2 when n β‘2(mod 4). f(ei) = 0 for 2t+k<i β€n Note that The number distribution is 1) for nβ‘1(mod 4) vf(0,1)=( 2+3, 2+4), ef(0,1)=( 2+4, 2+4) π π π π 2) for nβ‘2(mod 4) vf(0,1)=( 2+4, 2+4), ef(0,1)=( 2+5, 2+4) π π π π 3) for nβ‘3(mod 4) vf(0,1)=( 2+5, 2+4), ef(0,1)=( 2+5, 2+5) π π π π Note that The number distribution is 1) for nβ‘1(mod 4)vf(0, 1)=( 2+3, 2+4), ef(0,1)=( 2+4, 2+4) π π π π 2) for nβ‘2(mod 4)vf(0, 1)=( 2+4, 2+4), ef(0,1)=( 2+5, 2+4) π π π π 3) for nβ‘3(mod 4)vf(0, 1)=( 2+5, 2+4), ef(0,1)=( 2+5, Theorem K4(Pn) is E-cordial. Proof: The K4(Pn) is defined as V(G)={ u1, u2, u3, u4, u5, u6}U{v1, v2,.vn}and E(G) ={c1=(v1u1), c2=(u1u2), c3=(u2u3), c4=(u3u4), c5=(v1u2), c6=(u1u3)} U{c7=(vnu4), c8=(u4u5), c9=(u5u6), c10=(u6vn), c11=(vnu5), c12=(u5u6)} U{ei=(vivi+1) / I =1, 2β¦, n-1} We diagramatically show below the E-cordial labeling of K4(p2), K4(p3), K4(P5). 2+5)
Conclusions
We have discussed only for single path between two copies of graphs. It is necessary to evaluate E-cordiality if there are 2 or more paths between twocopies of graphs.
References
5 in the original paper- Bapat Mukund V. Ph. D. Thesis, University of Mumbai, 2004.
- Joe Gallian. Dynamic survey of graph labeling, 2016.
- Harary, Graph Theory. Narosa publishing, New Delhi.
- Yilmaz Cahit. E-cordial graphs, Ars combina, 46:251- 256.
- Mukund Bapat V, Hindale Devgad, Sindhudurg. Maharashtra India, 416630. πβπ Define a function f :E(G)ο {0,1} as f(c1)=1, f(c2)=1, f(c4)=1, f(c7)=1, f(c8)=1, f(c10)=1. take t = πβ1 4 /j=1,2,3. ( n is divisible by 4 then graph is not E-cordial) For i β€2t f(ei)=0 for i is odd number β€2t. f(ei)=1 for i is even number <2t. f(ei)=1 for i= 2t, 2t+1, 2t+k where k = p-t+1 where p = 2 for πβ2 n β‘1,3 (mod 4) and p = 2 when n β‘2(mod 4). f(ei) = 0 for 2t+k<i β€n