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

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a study of ic coloring of graphs

A Study of IC-coloring of Graphs

研 究 生:林耀仁

指導教授:江南波

sum saturable
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.

ic coloring
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}

1995 penrice 5
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
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
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.

slide7

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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

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p=6

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p=5

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p=8

slide13

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}.

slide14

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 :
slide18

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 :
slide22

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 :
slide26

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.

slide27

Conjecture 2.1.4.

Suppose that T is a tree of order p. If Δ(T) >

,then T is sum-saturable.

slide28

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.

slide29

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.

slide30

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

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h = 3

h = 4

slide32

Corollary 2.2.2.

A perfect complete n-ary tree is sum-saturable.

m h 1 1 n m h 2 n m h log 2 m h 1 log 2 n log 2 m h 1 i e log 2 m h 1 log 2 m h log 2 n 1 n
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=

slide34

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.

slide35

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:
slide36

(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.

slide37

V1

Vn

V2

Theorem 3.1.2.

For every integer n 4, we have 2n-5 M(Kn-P3) 2n-4.

slide38

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:
slide39

(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.

slide40

V2

Vn

V1

Vn-1

Theorem 3.2.1.

For every integer n 4, we have 2n-9 M(Kn-P4) 2n-6.

slide41

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.

slide42

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).

slide43

V1

Vn

Vn-1

Theorem 3.2.4.

For every integer n 4, we have 2n-7 M(Kn-C3) 2n-5.

slide44

V1

V2

Vn

V3

Theorem 3.2.5.

For every integer n 5, we have 2n-9 M(Kn-k1,3) 2n-8.

slide45

Corollary 3.2.6.

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

references
References

[1] B. Bolt. Mathematical Cavalcade, Cambridge University

Press, 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 with

Applications, Macmillan,North-Holland: New York, Amsterdam, Oxford.

[4] J.F. Fink, Labelings that realize connected subgraphs of

all conceivable values,

Congressus Numerantium, 132(1998), pp.29-37.

[5] S.G. Penrice, Some new graph labeling problem: a

preliminary report, DIMACS Tech. Rep. 95-26(1995), pp.1-9.

references1
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.

references2
References

[11] R. Alter, J.A. Barnett, A postage stamp problem, Amer.

Math. Monthly 87(1980), pp.206-210.

[12] R. Guy, The Postage Stamp Problem, Unsolved Problems in

Number Theory, second ed., Springer, New York, 1994, pp.123-127.

[13] R.L. Heimer, H. Langenbach, The stamp problem, J.

Recreational Math. 7 (1974), pp.235-250.

[14] W.F. Lunnon, A postage stamp problem, Comput. J.

12(1969), pp.377-380.