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

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.

)(

)(HVx

kxf

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}

)(

)(HVv

vf

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.

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).

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.

We display the results of the sum-saturability of all non-isomorphic trees with at most p=8 vertices

p=1 p=2

p=3

p=4

11 2

1 3 2

1 4 3

2

p=5

p=62 4 3 1

5

34 5 2

1

1 6 4 5 3

2

1 4 5 6

2 3

1

2 4 5 63

2 3 4

16

5

p=7

1 4 5 6 7

2 3

1 7 3 5 6

2 42 6 7 3 5 4

1

1

2 4 5 6 73

1

2 4 5 6 73

2 5 6 7

1

3 4

35 6 7

4

2 17

1

2

3 4

5

6

p=8

1 5 8

4

7 3 6 2

1 4 7 5

82

6 31 4 5 6 7 8

2 3

p=8

p=8

A T-graph be an undirected graph with p vertices consisting of vertex set V(T) = {V1, V2, … , Vp} and edge set E(T) = {V1V2, V2V3, V3V4, … , Vp-

2Vp-1, VpV2}.

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 :

)(

)(HVx

kxf

Case 1.

Case 2.

Case 3.

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 :

)(

)(HVx

kxf

Case 1.

Case 2.

Case 3.

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 :

)(

)(HVx

kxf

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.

Conjecture 2.1.4. Suppose that T is a tree of order p. If

Δ(T) > ,then T is sum-saturable.

2p

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.

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.

If h ≧3, we define a labelling as follows ：

h21 2 4 8

92 1 h

3

5

h = 3

h = 4

1 2 4 8

73

5

6

9

10

11

12

13

14 15

1 2 4 8 16

23

3

5

6

7

9

10

11

12

13

14

15

1718

19 20

21

22

24

25

26

27

28

29

30

31

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

saturable.

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

L=

1 2

p

2L

p2log

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.

V1

V2

Vn-1

Vn

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:

2

1

12n

i

i

(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.

Theorem 3.1.2. For every integer n 4, we have 2n-

5 M(Kn-P3) 2n-4.

V1

V2Vn

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:

1

1

12n

i

i

(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.

Theorem 3.2.1. For every integer n 4, we have 2n-

9 M(Kn-P4) 2n-6.

V1

V2

Vn-1Vn

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.

V1

V2

V3 Vn-2

Vn-1

Vn

Theorem 3.2.3. For every integer n 5, we have 2n-

12 M(Kn-P3 {e}) 2n-5, where e E(P3).

V1

V2

V3 Vn-1Vn

Theorem 3.2.4. For every integer n 4, we have 2n-

7 M(Kn-C3) 2n-5.

V1

Vn Vn-1

Theorem 3.2.5. For every integer n 5, we have 2n-

9 M(Kn-k1,3) 2n-8.

V1 V2

V3Vn

Corollary 3.2.6. For every integer n m+1, we have

2n-2m-1 M(Kn-k1,m) 2n-2m.

References[1] B. Bolt. Mathematical Cavalcade, Cambridge UniversityPress, Cambridge, 1992.[2] Douglas. B. West (2001), Introduction to Graph Theory,Upper Saddle River, NJ 07458: Prentice Hall.[3] J. A. Bondy and U. S. R. Murty(1976), Graph Theory

withApplications, Macmillan,North-Holland: New York,

Amsterdam, Oxford.[4] J.F. Fink, Labelings that realize connected subgraphs ofall conceivable values, Congressus Numerantium, 132(1998), pp.29-37.[5] S.G. Penrice, Some new graph labeling problem: apreliminary report, DIMACS Tech. Rep. 95-26(1995), pp.1-

9.

References[6] E. Salehi, S. Lee and M. Khatirinejad, IC-Colorings andIC-Indices of graphs, Discrete Mathematics, 299(2005), pp.

297-310. [7] Chin-Lin Shiue, The IC-Indices of Complete BipartiteGraphs, preprint.[8] Nam-Po Chiang and Shi-Zuo Lin, IC-Colorings of the Joinand the Combination of graphs,Master of Science Institute

of AppliedMathematics Tatung University, July 2007.[9] Bao-Gen Xu, On IC-colorings of Connected Graphs,

Journalof East China Jiaotong University, Vol.23 No.1 Feb, 2006.[10] Hung-Lin Fu and Chun-Chuan Chou, A Study of StampProblem, Department of Applied Mathematics College of

ScienceNational Chiao Tung University , June 2007.

References

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Math. Monthly 87(1980), pp.206-210.[12] R. Guy, The Postage Stamp Problem,

Unsolved Problems inNumber Theory, second ed., Springer, New

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problem, J.Recreational Math. 7 (1974), pp.235-250.[14] W.F. Lunnon, A postage stamp problem,

Comput. J.12(1969), pp.377-380.