IC-Colorings and IC-Indices of graphs

Salehi, Ebrahim; Lee, Sin-Min; Khatirinejad, Mahdad

doi:10.1016/j.disc.2004.02.026

Cited by 5 publications

(18 citation statements)

References 11 publications

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“…ðf ðv 1 Þ; f ðv 2 ÞÞ is equivalent to ðf ðv 2 Þ; f ðv 1 ÞÞ. For instance, (1, 2) is equivalent to (2,1); both indicate that two adjacent vertices v 1 and v 2 exist where f ðv 1 Þ ¼ 1 and f ðv 2 Þ ¼ 2.…”

Section: Notationsmentioning

confidence: 98%

“…1, the maximal IC-colorings are also ideal with MðC n Þ ¼ IðC n Þ for 3 6 n 6. Without providing a systematic way to generate maximal IC-colorings of cycle, Salehi et al [1] indicated that C 7 does not have any ideal IC-coloring. By exhaustive search, 13 maximal IC-colorings of C 7 are discovered.…”

Section: Introductionmentioning

confidence: 98%

“…; Sðf Þg, where Sðf Þ is the maximum number that can be produced by f. The IC-index MðGÞ of the graph G is the number maxfSðgÞjg is an IC-coloring of Gg. The number of connected subgraph of a graph G is the natural upper bound of IC-index as described by Salehi et al [1]. An IC-coloring f of G is maximal if it is an IC-coloring of G when Sðf Þ ¼ MðGÞ.…”

Section: Introductionmentioning

confidence: 99%

“…An IC-coloring f is ideal if Sðf Þ is equal to the number of connected induced subgraph of G. Saheli et al [1] introduced the problem of finding IC-indices and IC-colorings of finite graphs. This problem may be considered as a derivative of the postage stamp problem in number theory [2][3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

“…This problem may be considered as a derivative of the postage stamp problem in number theory [2][3][4][5][6][7][8]. Salehi et al [1] have also studied the IC-indices of complete graphs, stars, doublestarts, paths, cycles, and wheels. Shiue and Fu [9] obtained the IC-index of a complete bipartite graph, K m;n , with MðK m;n Þ ¼ 3 Á 2 mþnÀ2 À 2 mÀ2 þ 2, for 2 6 m 6 n. Liu and Lee [10,11] investigated the IC-colorings and IC-indices of complete d-partite graphs and provided some properties and initial results.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A parallel algorithm for generating ideal IC-colorings of cycles

Liu

2015

Applied Mathematics and Computation

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Section: Notationsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A parallel algorithm for generating ideal IC-colorings of cycles

Liu

2015

Applied Mathematics and Computation

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CogDFH- a Cognitive-Based differential frequency hopping network

Cheng

Shu

Xiaoxu

et al. 2009

MILCOM 2009 - 2009 IEEE Military Communications Conference

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The IC-indices of Complete Multipartite Graphs

Shiue¹,

Lu²,

Kuo³

2017

Taiwanese J. Math.

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A coloring of a connected graph G is a function f mapping the vertex set of G into the set of all integers. For any subgraph H of G, we denote the sum of the values of f on the vertices of H as f (H). If for any integer k ∈ {1, 2, · · · , f (G)}, there exists an induced connected subgraph H of G such that f (H) = k, then the coloring f is called an IC-coloring of G. The IC-index of G, denoted as M (G), is the maximum value of f (G) over all possible IC-colorings f of G. In this paper, we present a useful method from which a lower bound on the IC-index of any complete multipartite graph can be derived. Subsequently, we show that, for m ≥ 2 and n ≥ 2, our lower bound on M (K 1(n),m ) is the exact value of it. . Research supported in part by NSC-102-2115-M-239-002.for 2 ≤ m ≤ n, in 2008, the problem regarding complete bipartite graphs was completely settled. In this present paper, we deal with complete multipartite graphs. A complete multipartite graph K m 1 ,m 2 ,··· ,m k is a graph whose vertex set can be partitioned into k partite sets V 1 , V 2 , · · · , V k , where |V i | = m i for all i ∈ {1, 2, · · · , k}, such that there are no edges within each V i and any two vertices from different partite sets are adjacent. A complete multipartite graph with k partite sets is called a complete k-partite graph as well. We also denote as K 1(n),m n+1 ,m n+2 ,··· ,m k , n ≤ k, the complete k-partite graphs in which there are n partite sets which are of size one and the rest (k − n) partite sets have sizes m n+1 , m n+2 , · · · , and m k . Therefore K 1(n) represents the complete graph K n .In this paper, we introduce some useful lemmas in Section 2. In Section 3, our main results are presented. We start with a useful proposition which gives a lower bound on the IC-index of the join of an independent set and a given connected graph. Consequently, a lower bound on the IC-index of any complete mutlipartite graph can be deduced. We shall show that our lower bound on M(K 1(n),m ) is indeed the exact value of it for m ≥ 2 and n ≥ 2. A concluding remark will be given in Section 4.

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IC-Colorings and IC-Indices of graphs

Cited by 5 publications

References 11 publications

A parallel algorithm for generating ideal IC-colorings of cycles

A parallel algorithm for generating ideal IC-colorings of cycles

CogDFH- a Cognitive-Based differential frequency hopping network

The IC-indices of Complete Multipartite Graphs

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