graceful trees through graceful codes (1)

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  1. 1. 26VIT 1984-2010 Creating Stars
  2. 2. 26VIT 1984-2010 Creating Stars A Place to Learn ; A Chance To Grow
  3. 3. 26VIT 1984-2010 Creating Stars ncmowli@hotmail.com N.Chandramowliswaran Applied Algebra Division School of Advanced Sciences VIT University 26VIT 1984-2010 Creating Stars ncmowli@hotmail.com Claude Berge Bill Tutte Generation of Graceful Trees through Graceful Codes
  4. 4. Abstract Graceful Code is a way to represent graceful graph in terms of sequence of non-negative integers. Given a graceful graph G on q edges, we can generate its graceful code in the form of (a1, a2, a3, ., aq-1, aq=0) to represent the graph. Similarly, we can easily draw the graph from the given graceful code. Graceful codes are classified into two categories, namely, -valuable code and gracious code based on their properties. Graceful code provides an useful and efficient techniques to study and analyze graphs using computer. Here we discuss generation of infinitely many graceful codes, -valuable codes and gracious codes for a given graceful code, -valuable code and a gracious code.
  5. 5. Introduction A simple graph G(V,E) on p vertices and q edges is said to be graceful if there exist an injection f: V{0, 1, 2,.,q} such that the induced function g: E{1, 2, 3, , q} which is defined by g(u, v)=|f(u)-f(v)| for every edge (u, v), is a bijective function; then f is called graceful labelling of G.
  6. 6. Graceful Code Let G be any graceful graph on q edges then (a1, a2, a3, , aq -1, aq) is called a graceful code of G, if 0 ai q - i; 1 i q. Here ai is the lower end vertex of the edge label i. It is important to note that aq is always zero For every graceful graph G we can write its code. Conversely, for every given graceful code we can draw the corresponding graceful graph as follows. Join edges:(a1,1+a1),(a2,2+a2),...,(aq - 1,q-1+aq-1), (aq, q+aq)
  7. 7. Example 1 Figure 1 shows a graceful graph on q = 7 edges with edge labeled from 1 to 7. 7 0 6 3 12 4 5 Code = (4, 2, 3, 0, 1, 0, 0) Figure 1
  8. 8. - valuable Code A graceful code (a1, a2, a3,..., aq-1, aq) of a graceful graph G on q edges is called - valuable code if Here a1 is called the separator or critical value of the - valuable code. a1 ai Max{ai| 1 i q} < Min{i+ai| 1 i q}.
  9. 9. Proposition (a1, a2, a3, ,aq-1,aq) represents -valuable code if and only if 0 (a1 aq - i + 1) / q i) 1 for all i , 1 i q - 1 Equivalently (a1, a2, a3, ,aq-1,aq) represents an - valuable code if and only if (a1 - aq, a1 - aq -1, , a1 - a3, a1 - a2, 0) represents a code of a Graceful Graph.
  10. 10. Properties of Graceful Codes 1.1 If (a1, a2, a3,, aq - 1, aq) represents a code of a graceful graph G on q edges, then, (a2, a3,, aq - 1, aq) represents a code of some graceful graph H on q - 1 edges. 1.2 If (a1, a2, a3,, aq-1, aq) is an valuable code on q edges and (q -1 - a1 > a1) then (q 1 - a1, q 2 - a2, , 1- aq - 1, 0, a1, a2,a3,, aq) is an valuable code on 2q edges.
  11. 11. 1.3. If (a1, a2, a3,, aq - 1, aq) is an valuable code on q edges and (a1> q -1 - a1) then, (a1, a2, a3,, aq, q - 1- a1, q 2 - a2, ,1 aq - 1, 0) is an valuable code on 2q edges. 1.4. If (a1, a2, a3,, aq1 - 1, aq1) and (b1, b2, b3, , bq2 - 1, bq2) represents valuable codes on q1 and q2 edges respectively and a1 b1 then, (a1, a2, a3,, aq1 - 1, aq1, b1, b2, b3,, bq2 - 1, bq2) represents an valuable code on q1 + q2 edges.
  12. 12. 1.5. Let (a1, a2, a3,, aq -1, aq) represents a graceful code of a graph G on q edges then, (aq+ q, aq - 1+ q - 1,, 2 + a2, 1 + a1, a1, a2, a3,, aq - 1, aq) represents a valuable code on 2q edges.
  13. 13. Properties of Graceful Codes If (a1, a2, a3,, aq - 1, aq) represents a graceful code of a graceful graph G on q edges then, (aq+ q, aq 1 + q - 1,,2 + a2, 1 + a1, x, a1, a2, a3, aq - 1, aq), [0 x q] represents an valuable code on 2q + 1 edges.
  14. 14. If (a1, a2, , aq 1, aq) represent a code of a graceful graph G on q edges, Then, (q aq, q aq 1, , q a2, q a1, a1, a2, , aq 1, aq) represent a -valuable code on 2 q edges.
  15. 15. Properties of Graceful Codes Let X1, X2, X3, , Xr represent r valuable codes on edges qi (1 i r) having separators si respectively. Then, r-1 r-2 r-3 ( sj + Xr , sj + Xr-1 , sj + Xr-2 , s1+ s2+ X3, s1+ X2, X1) j=1 j=1 j=1 r always represent a valuable code on qj edges. j = 1
  16. 16. Tree Generation Theorems Let G be any simple graph on n vertices and q edges. Define a bipartite graph HG as follows: (vi, vj) E(G) (vi, vj) E(HG) and (vi, vj) E(HG). Join any vk V(G) V(HG), [1 k n] to vk V(HG). Here |V(HG)| = 2|V(G)| and |E(HG)| = 2 | E(G)| +1.
  17. 17. Tree Generation Theorems Moreover if G has a code (a1, a2, a3,, aq - 1, aq) then HG has an valuable code (aq+ q, aq-1+q - 1,, 2+a2, 1 + a1, x, a1, a2, a3, , aq - 1, aq) [0 x q]. If G happens to be a bipartite graph, then HG contains two copies of G together with an edge connecting vk to vk
  18. 18. Examples Code = (0, 1, 0, 0) G HG Code = (4, 3, 3, 1, 3, 0, 1, 0, 0)
  19. 19. Examples G
  20. 20. Construction of HG
  21. 21. ai i + ai i E (G) ai i + ai q+1+ai q+1+i+ai E(HG) q+1+i q+1-i (aq+q, ,ai+i, ,1+a1, x, a1, , ai, , aq) q+1+ai q+1+i+ai q+1-i q+1+i i i
  22. 22. Tree Generation Theorems Theorem If (a1, a2, a3,, aq -1, aq) represents a valuable code of some tree T . Then, (aq+q, aq - 1+q-1, ,2+a2, 1+a1, a1, a2, a3,, aq - 1, aq) represents a valuable code of a tree S on 2q edges such that E(S) = E(T) U E(T).
  23. 23. Tree Generation Theorems Theorem If (a1, a2, a3,, aq-1, aq) is an valuable code of a graceful graph G on q edges, then, (a1, a2, a3,, aq-1, aq) represents a tree if and only if (a2, a3,, aq -1, aq) represents a tree on q - 1 edges.
  24. 24. Tree Generation Theorems If (a1, a2, , aq-2, aq-1, aq ) represents a code of a graceful tree on q edges, then 1. (qk - 1, ka1, (q 1) k 1, ka2, , 2 k - 1, kaq - 1, 1k - 1, kaq) represent a tree code on kq edges (k 2). 2. (qk - 1, ka1+r, (q 1) k 1, ka2+r, , 2k - 1, kaq-1+r, 1k - 1, kaq+r, 0r) ;1 r k, k 2 represent a tree code on kq+r edges.
  25. 25. Corollary 1 If (a1, a2, , aq - 2, aq - 1, aq ) represents a code of a graceful tree on q edges, then (q, 2a1, q - 1, 2a2, q - 2, 2a3, , 2, 2aq - 1, 1, 2aq) represent a code of a graceful tree on 2q edges and (q, 2a1+1, q - 1, 2a2+1, q - 2, 2a3+1, , 2, 2aq - 1+1, 1, 2aq+1,0) represent a tree code on 2q+1 edges.
  26. 26. Corollary 2 If (a1, a2, , aq-2, aq-1, aq ) represents a code of a graceful tree on q edges, then (q+1, 2a1, q, 2a2, q - 1, 2a3, , 3, 2aq-1, 2, 2aq, 1, 0) represent a code of a graceful tree on 2q+2 edges and (q+1, 2a1+1, q, 2a2+1, q - 1, 2a3+1, , 3, 2aq - 1+1, 2, 2aq+1, 1, 0, 0) represent a tree code on 2q + 3 edges.
  27. 27. Tree Generation Theorems Using - valuable tree codes Theorem 1 If (a1, a2, , aq - 1, aq) represent a -valuable tree code on q edges, then, (aq+q, aq - 1+ q 1, , 2 + a2, 1+ a1, 1 + a1, a1, a1,a2, , aq - 1, aq) represent a -valuable tree code on 2q+2 edges.
  28. 28. Theorem 2 Let (a1, a2, , aq1 - 2, aq1 - 1, aq1) represents a - valuable tree code on q1 edges and (b1, b2, , bq2 - 2, bq2 - 1, bq2) represent a tree code on q2 edges. Then, 1. (a1 + b1 , a1 + b2, , a1 + bq2 - 2, a1+ bq2 - 1, a1+ bq2, a1, a2, , aq1 - 2, aq1 - 1, aq1 ) represent a tree code on q1 + q2 edges.
  29. 29. Tree Generation Theorems Using - valuable tree codes 2. (a1+ b1, a1+b2, , a1+ bq2 - 2, a1+bq2 - 1, a1+bq2, a1 aq1, a1 aq1 - 1,a1 aq1 - 2, , a1 a2, 0) represent a tree code on q1+ q2 edges. 3. (q1 1 a1 + b1, q1 1 a1+ b2, q1 1 a1+ b3, , q1 1 a1+ bq2 - 2, q1 1 a1+ bq2 - 1, q1 1 a1+ bq2, q1 1 a1, q1 2 a2, , 2 aq1 - 2, 1 aq1 - 1, 0) represent a tree code on q1+ q2 edges.
  30. 30. Tree Generation Theorems Using - valuable tree codes 4. (q1 1 a1+ b1, q1 1 a1+ b2, q1 1 a1+ b3 , , q1 1 a1+ bq2 - 2, q1 1 a1+ bq2 - 1, q1 1 a1+ bq2, q1 1 a1, q1 2 (a1 aq1 - 1), q1 3 (a1 aq1 - 2), , 1 (a1 a2), 0) represent a tree code on q1+ q2 edges. 5. (a1+ a2, a1+ a3, a1+ a4, , a1+ aq1 - 2, a1+ aq1 - 1, a1+ aq1, a1, a2, , aq1 - 2, aq1-1, aq1) represent a tree code on 2q 1 edges.
  31. 31. Tree Generation Theorems Using - valuable tree codes Corollary 1 Let X1, X2, X3, , Xr represent r valuable tree codes on edges qi (1 i r) having separators si respectively. Then r- 1 r- 2 r- 3 ( sj + Xr, sj + Xr - 1, sj + Xr - 2, , s1+ s2+ X3, s1+ X2, X1) j=1 j=1 j=1 r always represent a valuable tree code on qj edges. j=1

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