International Journal of Engineering and Science Invention (IJESI)

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International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online

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International Journal of Engineering and Science Invention (IJESI)

  1. 1. International Journal of Engineering Science Invention ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726 www.ijesi.org Volume 2 Issue 10ǁ October. 2013 ǁ PP.01-17 K-Even Edge-Graceful Labeling of Some Cycle Related Graphs 1 Dr. B. Gayathri, 2S. Kousalya Devi 1,2 M.Sc., M.Phil., Ph.D., Pgdca., M.C.A., Associate Professor And Research Advisor Post Graduate & Research Department Of Mathematics Periyar E.V.R.College (Autonomous) Tiruchirappalli-620 023. Tamilnadu. India. ABSTRACT: In 1985, Lo[6] introduced the notion of edge-graceful graphs. In [4], Gayathri et al., introduced the even edge-graceful graphs. In [8], Sin-Min Lee, Kuo-Jye Chen and Yung-Chin Wang introduced the k-edgegraceful graphs.We introduced k-even edge-graceful graphs. In this paper, we investigate the k-even edgegracefulness of some cycle related graphs. KEYWORDS: k-even edge-graceful labeling, k-even edge-graceful graphs. AMS MOS) subject classification: 05C78. I. INTRODUCTION All graphs in this paper are finite, simple and undirected. Terms not defined here are used in the sense of Harary[5]. The symbols V(G) and E(G) will denote the vertex set and edge set of a graph G. The cardinality of the vertex set is called the order of G denoted by p. The cardinality of the edge set is called the size of G denoted by q. A graph with p vertices and q edges is called a (p, q) graph. In 1985, Lo[6] introduced the notion of edge-graceful graphs. In [4], Gayathri et al., introduced the even edge-graceful graphs and further studied in. In [8], Sin-Min Lee, Kuo-Jye Chen and Yung-Chin Wang introduced the k-edge-graceful graphs. We have introduced k-even edge- graceful graphs. Definition 1.1: k-even edge-graceful labeling (k-EEGL) of a (p, q) graph G(V, E) is an injection f from E to {2k – 1, 2k,   2k + 1, ..., 2k + 2q – 2} such that the induced mapping f defined on V by f ( x) = (f(xy)) (mod 2s) taken over all edges xy are distinct and even where s = max{p, q} and k is an integer greater than or equal to 1. A graph G that admits k-even edge-graceful labeling is called a k-even edge-graceful graph (K-EEGG). Remark 1.2: 1-even edge-graceful labeling is an even edge-graceful labeling. The definition of k-edge-graceful and k-even edge-graceful are equivalent to one another in the case of trees. The edge-gracefulness and even edge-gracefulness of odd order trees are still open. The theory of 1-even edge-graceful is completely different from that of k- even edge-graceful. For example, tree of order 4 is 2-even edge-graceful but not 1-even edge-graceful. In this paper we investigate the k-even edge-gracefulness of some cycle related graphs. Throughout this paper, we assume that k is a positive integer greater than or equal to 1. 2. Prior Results: 1.Theorem : If a (p, q) graph G is k-even edge-graceful with all edges labeled with even numbers and p  q then G is k-edge-graceful. 2.Theorem : If a (p, q) graph G is k-even edge-graceful in which all edges are labeled with even numbers and p  q then q(q  2k  1)  Further q(q + 2k – 1)  p( p  1) (mod p) . 2 0(mod p) if p is odd  p  2 (mod p) if p is even  www.ijesi.org 1 | Page
  2. 2. K-Even Edge-Graceful Labeling Of… 3.Theorem : If a (p, q) graph G is k-even edge-graceful in which all edges are labeled with even numbers and p  q then p  0, 1 or 3 (mod 4). 4.Theorem : If a (p, q) graph G is a k-even edge-graceful tree of odd order then p (l – 1) + 1 2 k= where l is any odd positive integer and hence k  1(mod p). 5. Observation : We observe that any tree of odd order p has the sum of the labels congruent to 0 (mod p). 6. Theorem : If a (p, q) graph G is a k-even edge-graceful tree of even order with p  0 (mod 4) then k = k Further p (2l – 1) + 1 where l is any positive integer. 4 p4  4 (mod p) if l is odd    3 p  4 (mod p) if l is even  4  2. MAIN RESULTS Definition 2.1 Let Cn denote the cycle of length n. Then the join of {e} with any one vertex of Cn is denoted by Cn  {e}. In this graph, p = q = n + 1. Theorem 2.2 The kz graph Cn  {e} p p   mod  , where 0  z   1 , 2 2  of even order is for all n  1 (mod 4) and n  1. Proof Let {v1, v2, ..., vn, v} be the vertices of Cn  {e}, the edges ei = and en+1 = k-even-edge-graceful  v2 , v  (see Figure 1). vn-1  vi , vi1  for 1  i  n – 1; e =  vn , v1  n vn en-1 en ... v1 .. e1 . e2 v3 en+1 v2 v Figure 1: Cn  {e} with ordinary labeling First, we label the edges as follows: For k  1, 1  i  n and i is odd, f  ei  = 2k + i – 2. When i is even, we label the edges as follows: For p2 p  , k  z  mod  , 0  z  4 2  www.ijesi.org 2 | Page
  3. 3. K-Even Edge-Graceful Labeling Of… For For For n  4z  7  2k  n  i  2 for 1  i   2 f  ei  =  n  4z  7  2k  n  i for  i  n.  2  p2 p k  mod  , 4  2 f  ei  = 2k + n + i. k p6 p  mod  , 4  2 f  ei  = 2k + n + i – 2. p  p  10 p  k  z  mod  ,  z   1, 2 4 2  3n  4 z  9  2k  n  i  2 for 1  i   2 f  ei  =  3n  4 z  9  2k  n  i for  i  n.  2   2k when k  0  mod p  f  en1  =  2k  2n  2 z  2 when k  z  mod p  and 1 z  p  1. Then the induced vertex labels are as follows: p2 p   mod  , 0  z  4 2  n  4z  7  n  4 z  2i  5 for 1  i   2 f   vi  =   4 z  n  2i  5 for n  4 z  7  i  n.  2  p+2 p Case 2: k   mod  4  2  f  v1  = 2n f   vi  = 2i – 2 ; Case 1: k  z Case 3: k f   vi  Case 4: for 2  i  n. p+6 p  mod  4  2 = 2i for 1  i  n. p  p  10 p  k  z  mod  ,  z  1 2 4 2  www.ijesi.org 3 | Page
  4. 4. K-Even Edge-Graceful Labeling Of… 3n  4 z  9  4 z  n  2i  7 for 1  i <  2 f   vi  =  4 z  3n  2i  7 for 3n  4 z  9  i  n.  2  p p  For k  z  mod  , 0  z   1, 2 2  f   vn1  = 0. Therefore, labels Cn  0z are {e} of f  V  = {0, 2, 4, ..., 2s – 2}, where s = max{p, q} = n + 1. So, it follows that the vertex all even distinct order is and even. k-even-edge-graceful for Hence, all k the graph  p   mod  , 2  where z p  1 , n  1 (mod 4) and n  1.  2 For example, consider the graph C13  {e}. Here n = 13; s = max {p, q} = 14; 2s = 28. The 21-EEGL of C13  {e} is given in Figure 2. 51 65 67 6 4 49 53 8 2 10 41 26 56 12 61 0 55 24 14 47 22 16 20 59 43 18 57 45 Figure 2: 21-EEGL of C13  {e} For example, consider the graph C17  {e}. Here n = 17; s = max {p, q} = 18; 2s = 36. The 5-EEGL of C17  {e} is given in Figure 3. 23 41 21 28 26 30 43 32 25 24 39 34 22 19 9 20 2 36 37 0 29 18 4 17 16 35 6 14 15 8 12 10 33 11 31 13 Figure 3: 5-EEGL of C17  {e} www.ijesi.org 4 | Page
  5. 5. K-Even Edge-Graceful Labeling Of… Definition 2.3 [8] For p  4, a cycle (of order p) with one chord is a simple graph obtained from a p-cycle by adding a chord. Let the p-cycle be v1v2 ... vpv1. Without loss of generality, we assume that the chord joins v1 with any one vi, where 3  i  p – 1. This graph is denoted by Cp(i). For example cycle by q = p + 1. adding a chord between the C p  5 vertices means a graph obtained from a p- v1 and v5. In this graph, Theorem 2.4 The graph Cn(i), (n > 4), 3  i  n – 1, cycle with one chord of odd order is k-even-edge-graceful for all kz q q   mod  , where 0  z   1 . 2 2  Proof Let ei =  vi , vi1  {v1, v2, v3, ..., vi, for 1  i  n – 1; en = vi+1,  vn , v1  ..., vn} and en+1 = be the vertices of Cn(i), the edges  v1 , vi  , 3  i  n – 1 (see Figure 4). The chord connecting the vertex v1 with vi, (i  3) is shown in Figure 4. For this graph, p = n and q = n + 1. v1 vn en e1 v2 en-1 e2 v3 vn-1 e3 en-2 en+1 vn-2 v4 e4 vi+2 .. . . .. ei+2 ei+1 ei vi+1 vi Figure 4: Cn(i) with ordinary labeling First, we label the edges as follows: For k  1 and 1  i  n, f  ei  = f  en1  = when i is odd  2k  i  2  2k  n  i  2 when i is even.  2k when k  0  mod q   2k  2n  2 z  2 when k  z  mod q  , 1  z  q  1. Then the induced vertex labels are as follows: Case I: n  1 (mod 4) and n  1 q+2 q   mod  , 0  z  4 2  n  4z  7   4 z  n  2i  5 for 1  i   2 f   vi  =  4 z  n  2i  7 for n  4 z  7  i  n.  2  Case 1: k  z www.ijesi.org 5 | Page
  6. 6. K-Even Edge-Graceful Labeling Of… Case 2: k f   vi  q6 q  mod  4  2 = 2i for 1  i  n. q  q  10 q  k  z  mod  ,  z  1 2 4 2  3n  4 z  9  4 z  n  2i  7 for 1  i   2 f   vi  =  4 z  3n  2i  9 for 3n  4 z  9  i  n.  2  Case 3: Case II: n  3 (mod 4) and n  3 q q  k  z  mod  , 0  z  2 4  n  4z  7   4 z  n  2i  5 for 1  i   2 f   vi  =  4 z  n  2i  7 for n  4 z  7  i  n.  2  Case 1: Case 2: k f   vi  Case 3: q4 q  mod  4  2 = 2i – 2 for 1  i  n. q  q 8 q  k  z  mod  ,  z  1 2 4 2  f   vi  Therefore, = f  V  3n  4 z  9  4 z  n  2i  7 for 1  i    2  4 z  3n  2i  9 for 3n  4 z  9  i  n.  2   {0, 2, 4, ..., 2s – 2}, where s = max {p, q} = n + 1. So, it follows that the vertex labels are all distinct and even. Hence, the graph Cn(i), (n > 4), 3  i  n – 1, cycle with one chord of odd order is k-even-edge-graceful for all k  z 0z q  1.  2 For example, consider the graph q   mod  , 2  C9  5 . Here n = 9; s = max {p, q} = 10; 2s = 20. www.ijesi.org 6 | Page
  7. 7. K-Even Edge-Graceful Labeling Of… The 15-even-edge-graceful labeling of C9  5 is given in Figure 5. 6 29 37 2 8 39 45 10 0 40 31 35 12 18 41 43 33 14 16 Figure 5: 15-EEGL of C9(5) For example, consider the graph C11(6). Here n = 11; s = max {p, q} = 12; 2s = 24. The 11-EEGL of C11(6) is given in Figure 6. 0 4 31 21 6 41 22 33 29 8 24 20 23 39 18 10 35 27 16 12 25 37 14 Figure 6: 11-EEGL of C11(6) Theorem 2.5 The graph for all Cn  i  , (n  4), 3  i  n – 1, cycle with one chord of even order is k-even-edge-graceful k  z  mod q  , where 0  z  q – 1 and n  0  mod 4  . Proof Let the vertices and edges be defined as in Theorem 6.3.2. First, we label the edges as follows: For k  1, f  e1  = 2k + 1 f  ei  f  e2  = 2k – 1 = 2k + 2i – 3 ; for 3i  n  3. 2 n  3  i  n, 2 2k  2i  1 when i is odd f  ei  =  2k  2i  3 when i is even. For k  1 and www.ijesi.org 7 | Page
  8. 8. K-Even Edge-Graceful Labeling Of… f  en1  =  2k when k  0  mod q   2k  2n  2 z  2 when k  z  mod q  , 1  z  q  1. Then the induced vertex labels are as follows: Case 1: k  0 (mod q) f   v1  f   v3  = 2n – 2 ; f   v2  = 0. = 2. n  4i  8 for 4  i   3   2 f   vi  =  4i  2n  6 for n  3  i  n.  2  q 1 Case 2: k  z (mod q), 1  z  2  when i = 1, 2. f  vi  = 4z + 4i – 8 For k  z (mod q), 1  f   v3  z q 3 , 2 = 4z + 2 For 4  i  n, f   vi  For =  4 z  4i  8  4 z  4i  2n  10    4 z  4i  2n  6  4 z  4i  4n  8  n z3 2 n n for  z  3  i   3 2 2 n for  3  i  n  z  2 2 for n  z  2  i  n. for 4  i  q 1  mod q  , 2 f   v3  = 0 k n  4i  10 for 4  i   3   2 f   vi  =  4i  2n  8 for n  3  i  n.  2  q +1 Case 3: k   mod q  2 f   v1  = 2n f   v2  = 2. ; f   v3  = 4. www.ijesi.org 8 | Page
  9. 9. K-Even Edge-Graceful Labeling Of… f   vi  Case 4:  4i  6  = 0    4i  2n  4  k  z  mod q  , f   vi  f   v3  q+3  z  q 1 2 = 4z + 4i – 2n – 10 when i = 1, 2. = 4z – 2n. For k  z (mod q), f   vi  n 2 2 n when i   2 2 n for  3  i  n. 2 for 4  i < q3  z  q2, 2 4 z  4i  2n  10  4 z  4i  4n  12 =    4 z  4i  4n  8  4 z  4i  6n  10   for 4  i  n  z  3 n for n  z  3  i   3 2 n 3n for  3  i  z3 2 2 3n for  z  3  i  n. 2 For k  q – 1 (mod q), f   vi  n  4i  12 for 4  i   3   2 =  4i  2n  10 for n  3  i  n.  2  Therefore, f  V   0, 2, 4, ..., 2s  2 , where s = max{p, q} = n + 1. So, it follows that the Cn  i  , (n  4), 3  i  n – 1, cycle will one chord of vertex labels are all distinct and even. Hence, the graph even order is k-even-edge-graceful for all k  z (mod q), where 0  z  q – 1 and n  0 (mod 4).  For example, consider the graph C16(5), Here n = 16; s = max{p, q} = 17; 2s = 34. The 18-EEGL of C16(5) is given in Figure 7. 0 30 4 37 6 35 65 39 68 67 12 41 26 16 61 43 22 20 63 45 18 24 57 14 47 59 49 53 10 51 2 28 32 Figure 7: 18-EEGL of C16(5) www.ijesi.org 9 | Page
  10. 10. K-Even Edge-Graceful Labeling Of… Definition 2. 6 The crown Cn  K1 is the graph obtained from the cycle Cn by attaching pendant edge at each vertex of the cycle and is denoted by Cn . In this graph, p = q = 2n. Theorem 2.7 The crown graph where  Cn ,  n  3 of even order is k-even-edge-graceful for all p  k  z  mod  , 3  p  1 and n  0 (mod 3). 3 0 z Proof For this graph, p = q = 2n. Let v1, v2, v3, ..., vn and C  n ' ' ' v1' , v2 , v3 , ..., vn be the vertices and pendant vertices of respectively. v1' e1' v1 e1 en vn ' vn v2 e2' v3 e3' en' en-1 e2 vn-1 vn' 1 v2' ' en 1 v3' e3 en-2 ... .. v4 . e4' .. v4' . Figure 8: + Cn with ordinary labeling The edges are defined by ei =  vi , vi1  and ei' = for 1  i  n – 1 ; v , v  i ' i en =  vn , v1  for 1  i  n (see Figure 8). First, we label the edges as follows: For k  1, f  ei  f  e1'  f  ei'  for 1  i  n = 2k + 4i – 5 = 2k for 2  i  n. = 2k + 4(n – i + 1) Then the induced vertex labels are as follows: Case 1: p  k  0  mod  3  www.ijesi.org 10 | Page
  11. 11. K-Even Edge-Graceful Labeling Of… f   vi  = 4n  4i  10 when i  1, 2  for 3  i  n. 4i  10 p p  k  z  mod  , 1  z   1 and z is odd 3 3  5  3z  6 z  4i  10 for 1  i  n    2 f   vi  =  6 z  4n  4i  10 for n  5  3z  i  n.  2  p p  Case 3: k  z  mod  , 1  z   1 and z is even 3 3  6  3z  for 1  i  n  6 z  4i  10  2 f   vi  =  6 z  4n  4i  10 for n  6  3z  i  n.  2  Case 2: The pendant vertices will have the labels (mod 2p) of the edges incident on them. Clearly, the vertex  Cn ,  n  3 labels are all distinct and even. Hence, the crown graph for all p  k  z  mod  , 3  where 0 z For example, consider the graph graceful labeling of C6 of even order is k-even-edge-graceful p  1 and n  0 (mod 3).  3 C6 . Here p = q = 12; s = max{p, q} = 12; 2s = 24. The 2-even-edge- is given in Figure 9. 4 4 23 8 8 6 3 19 12 12 24 10 2 0 7 22 15 14 18 20 20 11 16 16 Figure 9: 2-EEGL of www.ijesi.org + C6 11 | Page
  12. 12. K-Even Edge-Graceful Labeling Of… Theorem 2.8 The crown graph  Cn ,  n  4  of even order is k-even-edge-graceful for all k  z  mod p  , where 0  z  p – 1, n  1 (mod 3) and n  1. Proof Let the vertices and edges be defined as in Theorem 6.4.2. The edge labels are also same as in Theorem 6.4.2. Then the induced vertex labels are as follows: Case 1: k  0 (mod p) f   vi  = 4n  4i  10 when i  1, 2  for 3  i  n. 4i  10 When z is odd, the induced vertex labels are given below: Case 2: k  z (mod p), 1  z  p +1 3 5  3z  6 z  4i  10 for 1  i  n    2 f   vi  =  6 z  4n  4i  10 for n  5  3z  i  n.  2  p+4 2p+2 Case 3: k  z (mod p), z 3 3 5  3z  6 z  4n  4i  10 for 1  i  2n  2  f   vi  =  6 z  8n  4i  10 for 2n  5  3z  i  n.  2  2p+5 Case 4: k  z (mod p),  z  p1 3 5  3z  6 z  8n  4i  10 for 1  i  3n+ 2  f   vi  =  6 z  12n  4i  10 for 3n  5  3z  i  n.  2  When z is even, the induced vertex labels are given below: Case 5: k  z  modp  , 1  z  f   vi  Case 6: = p +1 3 6  3z  6 z  4i  10 for 1  i  n    2  6 z  4n  4i  10 for n  6  3z  i  n.  2  k  z  mod p  , p+4 2p+2 z  3 3 www.ijesi.org 12 | Page
  13. 13. K-Even Edge-Graceful Labeling Of… 6  3z  6 z  4n  4i  10 for 1  i  2n  2  f   vi  =  6 z  8n  4i  10 for 2n  6  3z  i  n.  2  2p+5 Case 7: k  z  mod p  , z  p  1 3 6  3z  6 z  8n  4i  10 for 1  i  3n    2 f   vi  =  6 z  12n  4i  10 for 3n  6  3z  i  n.  2  The pendant vertices will have the labels (mod 2p) of the edges incident on them. Clearly, the vertex  Cn ,  n  4  labels are all distinct and even. Hence, the crown graph for all of even order is k-even-edge-graceful k  z  mod p  , where 0  z  p – 1, n  1 (mod 3) and n  1.  For example, consider the graph graceful labeling of C  7 C7 . Here p = q = 14; s = max {p, q} = 14; 2s = 28. The 5-even-edge- is given in Figure 10. 10 10 24 9 33 14 20 14 0 29 18 18 34 6 13 16 4 25 30 2 17 12 8 21 22 26 26 22 Figure 10: 5-EEGL of C 7+ Theorem 2.9 The crown graph  Cn , (n  5) of even order is k-even-edge-graceful for all k  z (mod p), where 0  z  p – 1, n  2 (mod 3) and n  2. Proof Let the vertices and edges be defined as in Theorem 2.7. The edge labels are also same as in Theorem 2.7. Then the induced vertex labels are as follows: Case 1: k  0 (mod p) www.ijesi.org 13 | Page
  14. 14. K-Even Edge-Graceful Labeling Of… f   vi  = 4n  4i  10 when i  1, 2  for 3  i  n. 4i  10 When z is odd, the induced vertex labels are given below: Case 2: k  z (mod p), 1  z  f   vi  5  3z  6 z  4i  10 for 1  i  n   2 =   6 z  4n  4i  10 for n + 5  3z  i  n.  2  Case 3: k  z (mod p), f   vi  p+5 3 z 2p +1 3  6 z  4n  4i  10 =    6 z  8n  4i  10   Case 4: k  z (mod p), f   vi  p+2 3 2p+4 3 for 1  i  2n  for 2n  5  3z 2 5  3z  i  n. 2 zp–1   6 z  8n  4i  10 =   6 z  12n  4i  10   for 1  i  3n  for 3n  5  3z 2 5  3z  i  n. 2 When z is even, the induced vertex labels are given below: Case 5: k  z (mod p), 1  z  f   vi  6  3z  6 z  4i  10 for 1  i  n    2 =  6 z  4n  4i  10 for n  6  3z  i  n.  2  Case 6: k  z (mod p), f   vi  f  vi  p+5 3 z 2p +1 3  6 z  4n  4i  10 =    6 z  8n  4i  10   Case 7: k  z (mod p),  p+2 3 2p+4 3 for 1  i  2n  for 2n  6  3z 2 6  3z  i  n. 2 zp–1  6 z  8n  4i  10  =   6 z  12n  4i  10   for 1  i  3n  for 3n  6  3z 2 6  3z  i  n. 2 www.ijesi.org 14 | Page
  15. 15. K-Even Edge-Graceful Labeling Of… The pendant vertices will have the labels (mod 2p) of the edges incident on them. Clearly, the vertex  Cn , (n  5) of even order is k-even-edge-graceful for labels are all distinct and even. Hence, the crown graph all k  z (mod p), where 0  z  p – 1, n  2 (mod 3) and n  2.  For example, consider the graph The 10-even-edge-graceful labeling of C5 . Here p = q = 10; s = max {p, q} = 10; 2s = 20. C5 is given in Figure 11. 0 20 4 14 35 16 19 24 36 18 10 23 31 6 2 27 28 32 12 8 Figure 11: 10-EEGL of Definition 2.10 A graph Hm G  + C5 is obtained from a graph G by replacing each edge with m parallel edges. Theorem 2.11 The graph Hn(Pn), (n  2) of k  z (mod p – 1), where 0  z  p – 2 and n is even. Proof even order Let {v1, v2, ..., vn} be the vertices of Hn(Pn). Let the edges eij of eij = (vi, vi+1) for 1  i  n – 1, 1  j  n (see Figure 12). v1 v2 v3 ... e21 e11 is k-even-edge-graceful H n  Pn  ... for all be defined by vn-1 vn en-11 e12 e22 en-12 e13 e23 en-13 ... ... ... e1n e2n en-1n Figure 12: Hn(Pn) with ordinary labeling www.ijesi.org 15 | Page
  16. 16. K-Even Edge-Graceful Labeling Of… For this graph, p = n; q = n(n – 1). First, we label the edges as follows: For k  1, for 1  i  n  1, i is odd and 1 j  n 2k  ni  2 j  5 f  eij  =  2k  2 j  n(i  1)  4 for 1  i  n  1, i is even and 1 j  n . Then the induced vertex labels are as follows: Case 1: k  0 (mod p – 1) f   v1  = 2n (n – 2). f   vi  = f   vn  = n (n – 4). n  2n  2i  9  for 2  i  4  for 4  i  n  1. n  2i  7  Case 2: k  z (mod p – 1), 1  z  p – 2 f   v1  Subcase (i):  = 2n (z – 1). k  1 (mod p – 1) f (vi ) = n(2i – 3) for 2  i  n – 1. f  (vn ) = n[n + 2(z – 2)]. Subcase (ii): p 2 n  2i  3  4n  z  1 for 2  i  n  2 z  3    n  2  i  2 z  n   5 for n  2 z  3  i  n  1.   k  z (mod p – 1), 2  z  f   vi  = f  (vn ) = n[n + 2(z – 2)]. Subcase (iii): p+2  z  p2 2 n  2  i  2 z  n   5 for 2  i  2n  2 z  2     n  2  i  2n  2 z   3 for 2n  2 z  2  i  n  1.    k  z (mod p – 1), f   vi  = f  (vn ) = n[2(z – 1) – n]. Therefore, f  V   {0, 2, 4, ..., 2s – 2}, where s = max {p, q} = n(n – 1). So, it follows that the vertex labels are all distinct and even. Hence, the graph Hn(Pn), (n  2) of even order is k-even-edge-graceful for all k  z (mod p – 1), where 0  z  p – 2 and n is even.  For example, consider the graph H8(P8). Here p = 8; q = 56; s = max {p, q} = 56; 2s = 112. The 6-EEGL of H8(P8) is given in Figure 13. www.ijesi.org 16 | Page
  17. 17. K-Even Edge-Graceful Labeling Of… 80 17 56 18 72 33 88 34 104 49 8 50 24 65 19 20 35 36 51 52 67 21 22 37 38 53 54 69 23 24 39 40 55 56 71 25 26 41 42 57 58 73 27 28 43 44 59 60 75 29 30 45 46 61 62 77 31 32 47 48 63 64 16 79 Figure 13: 6-EEGL of H8(P8) REFERENCES [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. G.S. Bloom and S.W. Golomb, Applications of numbered undirected graphs, Proc. IEEE, 65(1977) 562 – 570. G.S. Bloom and S.W. Golomb, Numbered complete graphs, unusual rulers, and assorted applications, in Theory and Applications of Graphs, Lecture Notes in Math., 642, Springer-Verlag, New York (1978) 53 – 65. J.A. Gallian, A dynamic survey of graph labeling, The electronic journal of Combinatorics 16 (2009), #DS6. B. Gayathri, M. Duraisamy, and M. Tamilselvi, Even edge graceful labeling of some cycle related graphs, Int. J. Math. Comput. Sci., 2 (2007) 179 – 187. F. Harary, “Graph Theory” – Addison Wesley, Reading Mass (1972). S. Lo, On edge-graceful labelings of graphs, Congr. Numer., 50 (1985) 231 – 241. W.C. Shiu, M.H. Ling, and R.M. Low, The edge-graceful spectra of connected bicyclic graphs without pendant, JCMCC, 66 (2008) 171 – 185. Sin-Min Lee, Kuo-Jye Chen and Yung-Chin Wang, On the edge-graceful spectra of cycles with one chord and dumbbell graphs, Congressus Numerantium, 170 (2004) 171 – 183. www.ijesi.org 17 | Page

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