A Study of IC-coloring of Graphs

1. A Study of IC-coloring of Graphs 研 究 生：林耀仁 指導教授：江南波

2. Sum-saturable Let G = (V, E) be an undirected graph with p vertices and let K = p(p+1)/2. Let f be a bijective function from V to {1,2,...,p}. Then f is said to be a saturating labelling of G if, given any k (1 k K), there exists a connected subgraph H of G such that . If a saturating labelling of G exists, then G is said to be sum-saturable.

3. IC-coloring let G=(V, E) be an undirected graph and let f be a function from V to N. For each subgraph H of G, we define fs(H) = .Then f is said to be a IC-coloring of G if, given any k ( 1 k fs(G)) there exists a connected subgraph H of G such that fs(H)=k. The IC-index of G is defined to be M(G) = max{fs(G) | f is an IC-coloring of G}

4. 1995, Penrice[5] Theorem 1.3.1. For the complete graph Kn, M(Kn)=2n-1. Theorem 1.3.2. For every n 4, M(Kn-e)=2n-3. Theorem 1.3.3. For all positive integers n 2, M(K1,n)=2n+2.

5. 2005, E. Salehi et.al.[6] Observation 1.3.5. If H is a subgraph of G, then M(H) M(G). Observation 1.3.6. If c(G) is the number of connected induced subgraph of G, then M(G) c(G).

6. 2007, Chin-Lin Shiue[7] Theorem 1.3.8. For any complete bipartite graph Km, n, 2 m n, M(Km, n)=3 ． 2m+n-2-2m-2+2.

7. 2 1 4 3 1 1 2 1 3 2 We display the results of the sum-saturability of all non-isomorphictrees with at most p=8 vertices p=1 p=2 p=3 p=4

8. p=6 5 1 4 5 2 2 4 3 1 3 2 2 3 1 6 4 5 3 1 4 5 6 3 1 5 6 2 4 5 6 2 3 4 1 p=5

9. 1 2 6 7 3 5 4 2 3 2 4 1 1 4 5 6 7 1 7 3 5 6 3 3 2 5 6 7 4 2 4 5 6 7 1 2 1 3 4 1 6 7 5 6 7 7 6 2 5 3 2 5 3 4 1 4 p=7

10. 4 1 5 8 7 3 6 2 2 3 2 8 1 4 5 6 7 8 1 4 7 5 6 3 p=8

11. p=8

12. p=8

13. A T-graph be an undirected graph with p vertices consistingof vertex set V(T) = {V1, V2, … , Vp} and edge setE(T) = {V1V2, V2V3, V3V4, … , Vp-2Vp-1,VpV2}.

14. Theorem 2.1.1. The T-graph with order p=6 is not Sum-saturable. proof. • Assume T-graph with order p=6 is sum-saturable, then there is a saturating labeling f of T-graph. • Given any k (1 k 21=K), there exists a connected subgraph H of T such that . • Because K-1 and K-2 exists a connected subgraph H of T. So, number 1 and number 2 are labeling in end-vertex. • Hence we have following three cases :

15. Case 1.

16. Case 2.

17. Case 3.

18. Theorem 2.1.2. The T-graph with order p=7 is not Sum-saturable. proof. • Assume T-graph with order p=7 is sum-saturable, then there is a saturating labeling f of T-graph. • we given any k (1 k 28=K), there exists a connected subgraph H of T such that . • Because K-1 and K-2 exists a connected subgraph H of T. So, number 1 and number 2 are labeling in end-vertex. • Hence we have following three cases :

19. Case 1.

20. Case 2.

21. Case 3.

22. Theorem 2.1.3. The T-graph with order p=8 is not Sum-saturable. proof. • Assume T-graph with order p=8 is sum-saturable, then there is a saturating labeling f of T-graph. • we given any k (1 k 36=K), there exists a connected subgraph H of T such that . • Because K-1 and K-2 exists a connected subgraph H of T. So, number 1 and number 2 are labeling in end-vertex. • Hence we have following three cases :

26. Remark. We use the same way in Theorem 2.1.3, and we got the T-graph of order p 9 is not sum-saturable.

27. Conjecture 2.1.4. Suppose that T is a tree of order p. If Δ(T) > ,then T is sum-saturable.

28. A rooted tree T is a complete n-ary tree, if each vertex in T except the leaves has exactly n children. For each vertex v in T, if the length of the path from the root to v is L, then we say that v is on the level L. If all leaves are on the same level h, then we call T a perfect complete n-ary tree with the hight h.

29. Theorem 2.2.1. A perfect complete binary tree T is sum-saturable. Proof. Let h be the hight of T. If h 2 Hence perfect complete binary trees with height h 2 are sum-saturable.

30. 3 5 1 2 4 8 If h ≧3, we define a labelling as follows：

31. 6 3 9 7 5 10 13 15 1 2 4 8 11 12 14 14 6 15 3 7 25 17 23 5 9 11 18 21 26 29 1 2 4 8 16 10 12 13 19 20 22 24 27 28 30 31 h = 3 h = 4

32. Corollary 2.2.2. A perfect complete n-ary tree is sum-saturable.

33. m(h+1) = 1 + n．m(h) < 2．n． m(h) log2m(h+1) < log2n+ log2m(h)+1,i.e. log2m(h+1)- log2m(h) < log2n +1< n p 1 2 2L L=

34. Vn-1 V1 Vn V2 Theorem 3.1.1. For every integer n 4, we have 2n-8 M(Kn-L) 2n-3, where L is a matching consisting of two edges.

35. Proof. • Let V(Kn-L)={V1, V2, … , Vn}. • We assign the vertexV1 is non-adjacent to the vertex Vn-1 and the vertex V2 is non-adjacent to the vertex Vn. • We define f:V(Kn-L) N by f(Vi)=2i-1, for all i=1, 2, … ,n-2, f(Vn-1)=2n-2-2 and f(Vn)=2n-1-5. • We claim that f is an IC-coloring of V(Kn-L), with fs(Kn-L)= +(2n-2-2)+(2n-1-5)=2n-8. • For any integer k [1, 2n-8], and consider the following three cases:

36. (i) k [1, 2n-2-1] (ii) k [2n-2, 2n-1-3] Let a=k-(2n-2-2), then 2 a 2n-2 -1. (iii) k [2n-1-2, 2n-8] Let b=k-(2n-1 -5), then 3 b 2n-1 -3. We get 2n-8 M(Kn-L) 2n-3.

37. V1 Vn V2 Theorem 3.1.2. For every integer n 4, we have 2n-5 M(Kn-P3) 2n-4.

38. Proof. • Let V(Kn-P3)={V1, V2, … , Vn}. • We assign the vertexV1 is non-adjacent to the vertex Vn and the vertex V2 is non-adjacent to the vertex Vn. • We define f:V(Kn-P3) N by f(Vi)=2i-1, for all i=1, 2, … ,n-1, and f(Vn)=2n-1-4. • We claim that f is an IC-coloring of V(Kn-P3), with fs(Kn-P3)= +(2n-1-4)=2n-5. • For any integer k [1, 2n-5], and consider the following two cases:

39. (i) k [1, 2n-1 -1] (ii) k [2n-1, 2n-5] Let a=k-(2n-1-4), then 4 a 2n-1 -1. We get 2n-5 M(Kn-P3) 2n-4.

40. V2 Vn V1 Vn-1 Theorem 3.2.1. For every integer n 4, we have 2n-9 M(Kn-P4) 2n-6.

41. V2 Vn-1 Vn V3 Vn-2 V1 Theorem 3.2.2. For every integer n 6, we have 2n-20 M(Kn-R) 2n-4, where R is a matching consisting of three edges.

42. V2 Vn V3 Vn-1 V1 Theorem 3.2.3. For every integer n 5, we have 2n-12 M(Kn-P3 {e}) 2n-5, where e E(P3).

43. V1 Vn Vn-1 Theorem 3.2.4. For every integer n 4, we have 2n-7 M(Kn-C3) 2n-5.

44. V1 V2 Vn V3 Theorem 3.2.5. For every integer n 5, we have 2n-9 M(Kn-k1,3) 2n-8.

45. Corollary 3.2.6. For every integer n m+1, we have 2n-2m-1 M(Kn-k1,m) 2n-2m.

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47. References [6] E. Salehi, S. Lee and M. Khatirinejad, IC-Colorings and IC-Indices of graphs, Discrete Mathematics, 299(2005), pp. 297-310. [7] Chin-Lin Shiue, The IC-Indices of Complete Bipartite Graphs, preprint. [8] Nam-Po Chiang and Shi-Zuo Lin, IC-Colorings of the Join and the Combination of graphs,Master of Science Institute of Applied Mathematics Tatung University, July 2007. [9] Bao-Gen Xu, On IC-colorings of Connected Graphs, Journal of East China Jiaotong University, Vol.23 No.1 Feb, 2006. [10] Hung-Lin Fu and Chun-Chuan Chou, A Study of Stamp Problem, Department of Applied Mathematics College of Science National Chiao Tung University , June 2007.

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