ResearchArticle On the Locating Chromatic Number …downloads.hindawi.com/journals/ijmms/2018/5327504.pdftrees with locating chromatic number 3.In[]a lltrees of order with locating

Research ArticleOn the Locating Chromatic Number of Certain Barbell Graphs

Asmiati 1 I Ketut Sadha Gunce Yana1 and Lyra Yulianti2

1Mathematics Department Faculty of Mathematics and Natural Sciences Lampung UniversityJl Brodjonegoro No1 Bandar Lampung Indonesia2Mathematics Department Faculty of Mathematics and Natural Sciences Andalas UniversityKampus UNAND Limau Manis Padang 25163 Indonesia

Correspondence should be addressed to Asmiati asmiati308yahoocom

Received 27 March 2018 Revised 26 June 2018 Accepted 22 July 2018 Published 5 August 2018

Academic Editor Dalibor Froncek

Copyright copy 2018 Asmiati et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The locating chromatic number of a graph 119866 is defined as the cardinality of a minimum resolving partition of the vertex set 119881(119866)such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in 119866 are not containedin the same partition class In this case the coordinate of a vertex V in 119866 is expressed in terms of the distances of V to all partitionclasses This concept is a special case of the graph partition dimension notion In this paper we investigate the locating chromaticnumber for two families of barbell graphs

1 Introduction

The partition dimension was introduced by Chartrand et al[1] as the development of the concept of metric dimensionThe application of metric dimension plays a role in roboticnavigation [2] the optimization of threat detecting sensors[3] and chemical data classification [4] The concept oflocating chromatic number is a marriage between the parti-tion dimension and coloring of a graph first introduced byChartrand et al in 2002 [5] The locating chromatic numberof a graph is a newly interesting topic to study because thereis no general theorem for determining the locating chromaticnumber of any graph

Let 119866 = (119881 119864) be a connected graph We define thedistance as theminimum length of path connecting vertices 119906and V in119866 denoted by 119889(119906 V) A 119896-coloring of119866 is a function119888 119881(119866) 997888rarr 1 2 119896 where 119888(119906) = 119888(V) for any twoadjacent vertices 119906 and V in 119866 Thus the coloring 119888 inducesa partition Π of 119881(119866) into 119896 color classes (independent sets)1198621 1198622 119862119896 where 119862119894 is the set of all vertices colored bythe color 119894 for 1 le 119894 le 119896 The color code 119888Π(V) of a vertex V in119866 is defined as the 119896-vector (119889(V 1198621) 119889(V 1198622) 119889(V 119862119896))where 119889(V 119862119894) = min119889(V 119909) 119909 isin 119862119894 for 1 le 119894 le 119896 The119896-coloring 119888 of 119866 such that all vertices have different colorcodes is called a locating coloring of 119866 The locating chromatic

number of 119866 denoted by 120594119871(119866) is the minimum 119896 such that119866 has a locating coloring

The following theorem is a basic theorem proved byChartrand et al [5] The neighborhood of vertex 119906 in aconnected graph 119866 denoted by 119873(119906) is the set of verticesadjacent to 119906

Theorem 1 (see [5]) Let 119888 be a locating coloring in a connectedgraph 119866 If 119906 and V are distinct vertices of 119866 such that 119889(119906 119905) =119889(V 119905) for all 119905 isin 119881(119866)minus119906 V then 119888(119906) = 119888(V) In particular if119906 and V are non-adjacent vertices of 119866 such that119873(119906) = 119873(V)then 119888(119906) = 119888(V)

The following corollary gives the lower bound of thelocating chromatic number for every connected graph 119866

Corollary 2 (see [5]) If 119866 is a connected graph and there is avertex adjacent to 119896 leaves then 120594119871(119866) ge 119896 + 1

There are some interesting results related to the determi-nation of the locating chromatic number of some graphsTheresults are obtained by focusing on certain families of graphsChartrand et al in [5] have determined all graphs of order119899 with locating chromatic number 119899 namely a completemultipartite graph of 119899 vertices Moreover Chartrand et

HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2018 Article ID 5327504 5 pageshttpsdoiorg10115520185327504

2 International Journal of Mathematics and Mathematical Sciences

al [6] have succeeded in constructing tree on 119899 vertices119899 ge 5 with locating chromatic numbers varying from 3to 119899 except for (119899 minus 1) Then Behtoei and Omoomi [7]have obtained the locating chromatic number of the Knesergraphs Recently Asmiati et al [8] obtained the locatingchromatic number of the generalized Petersen graph 119875(119899 1)for 119899 ge 3 Baskoro and Asmiati [9] have characterized alltrees with locating chromatic number 3 In [10] all treesof order 119899 with locating chromatic number 119899 minus 1 werecharacterized for any integers 119899 and 119905 where 119899 gt 119905 + 3and 2 le 119905 lt 1198992 Asmiati et al in [11] have succeeded indetermining the locating chromatic number of homogeneousamalgamation of stars and their monotonicity properties andin [12] for firecracker graphs Next Wellyyanti et al [13]determined the locating chromatic number for complete 119899-ary trees

The generalized Petersen graph 119875(119899119898) 119899 ge 3 and 1 le119898 le lfloor(119899 minus 1)2rfloor consists of an outer 119899-cycle 1199101 1199102 119910119899a set of 119899 spokes 119910119894119909119894 1 le 119894 le 119899 and 119899 edges 119909119894119909119894+1198981 le 119894 le 119899 with indices taken modulo 119899 The generalizedPetersen graphwas introduced byWatkins in [14] Let us notethat the generalized Petersen graph 119875(119899 1) is a prism definedas Cartesian product of a cycle 119862119899 and a path 1198752

Next theorems give the locating chromatic numbers forcomplete graph 119870119899 and generalized Petersen graph 119875(119899 1)

Theorem3 (see [6]) For 119899 ge 2 the locating chromatic numberof complete graph 119870119899 is 119899

Theorem 4 (see [8]) The locating chromatic number ofgeneralized Petersen graph 119875(119899 1) is 4 for odd 119899 ge 3 or 5 foreven 119899 ge 4

The barbell graph is constructed by connecting twoarbitrary connected graphs119866 and119867 by a bridge In this paperfirstly we discuss the locating chromatic number for barbellgraph 119861119898119899 for 119898 119899 ge 3 where 119866 and 119867 are complete graphson119898 and 119899 vertices respectively Secondly we determine thelocating chromatic number of barbell graph 119861119875(1198991) for 119899 ge 3where 119866 and119867 are two isomorphic copies of the generalizedPetersen graph 119875(119899 1)

2 Results and Discussion

Next theoremproves the exact value of the locating chromaticnumber for barbell graph 119861119899119899

Theorem 5 Let 119861119899119899 be a barbell graph for 119899 ge 3 Then thelocating chromatic number of 119861119899119899 is 120594119871(119861119899119899) = 119899 + 1

Proof Let 119861119899119899 119899 ge 3 be the barbell graph with the vertexset 119881(119861119899119899) = 119906119894 V119894 1 le 119894 le 119899 and the edge set 119864(119861119899119899)= ⋃119899minus1119894=1 119906119894119906119894+119895 1 le 119895 le 119899 minus 119894 cup ⋃119899minus1119894=1 V119894V119894+119895 1 le 119895 le119899 minus 119894 cup 119906119899V119899

First we determine the lower bound of the locatingchromatic number for barbell graph 119861119899119899 for 119899 ge 3 Sincethe barbell graph 119861119899119899 contains two isomorphic copies of acomplete graph 119870119899 then with respect to Theorem 3 we have120594119871(119861119899119899) ge 119899 Next suppose that 119888 is a locating coloring

using 119899 colors It is easy to see that the barbell graph 119861119899119899contains two vertices with the same color codes which is acontradiction Thus we have that 120594119871(119861119899119899) ge 119899 + 1

To show that 119899 + 1 is an upper bound for the locatingchromatic number of barbell graph 119861119899119899 it suffices to provethe existence of an optimal locating coloring 119888 119881(119861119899119899) 997888rarr1 2 119899 + 1 For 119899 ge 3 we construct the function 119888 in thefollowing way

119888 (119906119894) = 119894 1 le 119894 le 119899

119888 (V119894) =

119899 for 119894 = 1

119894 for 2 le 119894 le 119899 minus 1

119899 + 1 otherwise

(1)

By using the coloring 119888 we obtain the color codes of 119881(119861119899119899)as follows

119888Π (119906119894)

=

0 for 119894119905ℎ component 1 le 119894 le 119899

2 for (119899 + 1)119905ℎ component 1 le 119894 le 119899 minus 1

1 otherwise

119888Π (V119894) =

0 for 119894119905ℎ component 2 le 119894 le 119899 minus 1

for 119899119905ℎ component 119894 = 1 and

for (119899 + 1)119905ℎ component 119894 = 119899

3 for 1119904119905 component 1 le 119894 le 119899 minus 1

2 for 1119904119905 component 119894 = 119899

1 otherwise

(2)

Since all vertices in 119881(119861119899119899) have distinct color codes thenthe coloring 119888 is desired locating coloring Thus 120594119871(119861119899119899) =119899 + 1

Corollary 6 For 119899119898 ge 3 and 119898 = 119899 the locating chromaticnumber of barbell graph 119861119898119899 is

120594119871 (119861119898119899) = max 119898 119899 (3)

Next theorem provides the exact value of the locatingchromatic number for barbell graph 119861119875(1198991)

Theorem 7 Let 119861119875(1198991) be a barbell graph for 119899 ge 3 Then thelocating chromatic number of 119861119875(1198991) is

International Journal of Mathematics and Mathematical Sciences 3

120594119871 (119861119875(1198991)) =

4 for odd 119899

5 for even 119899(4)

Proof Let 119861119875(1198991) 119899 ge 3 be the barbell graph with the vertexset 119881(119861119875(1198991)) = 119906119894 119906119899+119894 119908119894 119908119899+119894 1 le 119894 le 119899 and the edge set119864(119861119875(1198991)) = 119906119894119906119894+1 119906119899+119894119906119899+119894+1 119908119894119908119894+1 119908119899+119894119908119899+119894+1 1 le 119894 le119899minus 1 cup 1199061198991199061 1199062119899119906119899+1 1199081198991199081 1199082119899119908119899+1 cup 119906119894119906119899+119894 119908119894119908119899+119894 1 le119894 le 119899 cup 119906119899119908119899

Let us distinguish two cases

Case 1 (119899 odd) According to Theorem 4 for 119899 odd we have120594119871(119861119875(1198991)) ge 4 To show that 4 is an upper bound for thelocating chromatic number of the barbell graph 119861119875(1198991) wedescribe an locating coloring 119888 using 4 colors as follows

119888 (119906119894) =

1 for 119894 = 1

3 for even 119894 119894 ge 2

4 for odd 119894 119894 ge 3

119888 (119906119899+119894) =

2 for 119894 = 1

3 for odd 119894 119894 ge 3

4 for even 119894 119894 ge 2

119888 (119908119894) =

1 for odd 119894 119894 le 119899 minus 2

2 for even 119894 119894 le 119899 minus 1

3 for 119894 = 119899

119888 (119908119899+119894) =

1 for even 119894 119894 le 119899 minus 1

2 for odd 119894 119894 le 119899 minus 2

4 for 119894 = 119899

(5)

For 119899 odd the color codes of 119881(119861119875(1198991)) are

119888Π (119906119894)

=

119894 for 2119899119889 component 119894 le 119899 + 12

119894 minus 1 for 1119904119905 component 119894 le 119899 + 12

119899 minus 119894 + 1 for 1119904119905 component 119894 gt 119899 + 12


0 for 3119905ℎ component 119894 even 119894 ge 2

for 4119905ℎ component 119894 odd 119894 ge 3

1 otherwise

119888Π (119906119899+119894)

=

119894 for 1119904119905 component 119894 le 119899 + 12






1 otherwise

119888Π (119908119894)

=

119894 for 3119905ℎ component 119894 le 119899 minus 12

119894 + 1 for 4119905ℎ component 119894 le 119899 minus 12

119899 minus 119894 for 3119905ℎ component 119894 ge 119899 + 12

119899 minus 119894 + 1 for 4119905ℎ component 119894 ge 119899 + 12

0 for 2119899119889 component 119894 even 119894 le 119899 minus 1

for 1119904119905 component 119894 odd 119894 le 119899 minus 2

1 otherwise

119888Π (119908119899+119894)

=







1 otherwise(6)

Since all vertices in 119861119875(1198991) have distinct color codes then thecoloring 119888 with 4 colors is an optimal locating coloring and itproves that 120594119871(119861119875(1198991)) le 4

Case 2 (119899 even) In view of the lower bound fromTheorem 7it suffices to prove the existence of a locating coloring 119888 119881(119861119875(1198991)) 997888rarr 1 2 5 such that all vertices in 119861119875(1198991)have distinct color codes For 119899 even 119899 ge 4 we describe thelocating coloring in the following way

119888 (119906119894) =

1 for 119894 = 1

3 for even 119894 2 le 119894 le 119899 minus 2

4 for odd 119894 3 le 119894 le 119899 minus 1

5 for 119894 = 119899


119888 (119906119899+119894) =

2 for 119894 = 1

3 for odd 119894 119894 ge 3

4 for even 119894 119894 ge 2

119888 (119908119894) =

1 for odd 119894 119894 le 119899 minus 3

2 for even 119894 119894 le 119899 minus 2

3 for 119894 = 119899 minus 1

4 for 119894 = 119899

119888 (119908119899+119894) =

1 for even 119894 119894 le 119899 minus 2

2 for odd 119894 119894 le 119899 minus 1

5 for 119894 = 119899(7)

In fact our locating coloring of 119861119875(1198991) 119899 even has beenchosen in such a way that the color codes are

119888Π (119906119894)

=

119894 for 2119899119889 and 5119905ℎ components 119894 le 1198992

119894 minus 1 for 1119904119905 component 119894 le 1198992

119899 minus 119894 for 5119905ℎ component 119894 gt 1198992

119899 minus 119894 + 1 for 1119904119905 component 119894 gt 1198992


0 for 3119905ℎ component 119894 even 2 le 119894 le 119899 minus 2

for 4119905ℎ component 119894 odd 3 le 119894 le 119899 minus 1

2 for 4119905ℎ component 119894 = 1

for 3119905ℎ component 119894 = 119899

1 otherwise

119888Π (119906119899+119894)

=

119894 for 1119904119905 component 119894 le 1198992


119899 + 119894 for 5119905ℎ component 119894 le 1198992

119899 minus 119894 + 1 for 2119899119889 and 5119905ℎ components 119894 gt 1198992

119899 minus 119894 + 2 for 1119905ℎ component 119894 gt 1198992

0 for 3119905ℎ component 119894 odd 3 le 119894 le 119899 minus 1

for 4119905ℎ component 119894 even 2 le 119894 le 119899

2 for 3119905ℎ component 119894 = 1

1 otherwise

119888Π (119908119894)

=

119894 for 4119905ℎ component 119894 le 1198992


for 3119905ℎ component 119894 le 1198992 minus 1



119899 minus 119894 minus 1 for 3119905ℎ component 1198992 le 119894 le 119899 minus 1

0 for 1119904119905 component 119894 odd 119894 le 119899 minus 3

for 2119899119889 component 119894 even 119894 le 119899 minus 2

2 for 1119904119905 component 119894 = 119899 minus 1

for 2119899119889 component 119894 = 119899

1 otherwise

119888Π (119908119899+119894)

=




119899 minus 119894 for 3119905ℎ component 1198992 le 119894 le 119899 minus 1

for 5119905ℎ component 119894 gt 1198992




2 for 1119904119905 and 3119905ℎ components 119894 = 119899

1 otherwise(8)

Since for 119899 even all vertices of 119861119875(1198991) have distinct color codesthen our locating coloring has the required properties and120594119871(119861119875(1198991)) le 5 This concludes the proof

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Theauthors are thankful toDRPMDikti for the FundamentalGrant 2018


References

[1] G Chartrand P Zhang and E Salehi ldquoOn the partitiondimension of a graphrdquo Congressus Numerantium vol 130 pp157ndash168 1998

[2] V Saenpholphat and P Zhang ldquoConditional resolvability asurveyrdquo International Journal of Mathematics andMathematicalSciences vol 38 pp 1997ndash2017 2004

[3] M Johnson ldquoStructure-activity maps for visualizing the graphvariables arising in drug designrdquo Journal of BiopharmaceuticalStatistics vol 3 no 2 pp 203ndash236 1993

[4] G Chartrand and P Zhang ldquoTHE theory and applications ofresolvability in graphs A surveyrdquo vol 160 pp 47ndash68

[5] G Chartrand D Erwin M A Henning P J Slater and PZhang ldquoThe locating-chromatic number of a graphrdquo Bulletinof the Institute of Combinatorics and Its Applications vol 36 pp89ndash101 2002

[6] G Chartrand D Erwin M A Henning P J Slater and PZhang ldquoGraphs of order n-1rdquo Discrete Mathematics vol 269no 1-3 pp 65ndash79 2003

[7] A Behtoei and BOmoomi ldquoOn the locating chromatic numberof Kneser graphsrdquo Discrete Applied Mathematics The Journalof Combinatorial Algorithms Informatics and ComputationalSciences vol 159 no 18 pp 2214ndash2221 2011

[8] Asmiati Wamiliana Devriyadi and L Yulianti ldquoOn somepetersen graphs having locating chromatic number four or fiverdquoFar East Journal of Mathematical Sciences vol 102 no 4 pp769ndash778 2017

[9] E T Baskoro and Asmiati ldquoCharacterizing all trees withlocating-chromatic number 3rdquo Electronic Journal of GraphTheory and Applications EJGTA vol 1 no 2 pp 109ndash117 2013

[10] D K Syofyan E T Baskoro and H Assiyatun ldquoTrees withcertain locating-chromatic numberrdquo Journal of Mathematicaland Fundamental Sciences vol 48 no 1 pp 39ndash47 2016

[11] Asmiati H Assiyatun and E T Baskoro ldquoLocating-chromaticnumber of amalgamation of starsrdquo ITB Journal of Science vol43A no 1 pp 1ndash8 2011

[12] Asmiati H Assiyatun E T Baskoro D Suprijanto R Siman-juntak and S Uttunggadewa ldquoThe locating-chromatic numberof firecracker graphsrdquo Far East Journal of Mathematical Sciences(FJMS) vol 63 no 1 pp 11ndash23 2012

[13] D Welyyanti E T Baskoro R Simanjuntak and S Uttung-gadewa ldquoOn locating-chromatic number of complete n-arytreerdquo AKCE International Journal of Graphs and Combinatoricsvol 10 no 3 pp 309ndash315 2013

[14] M E Watkins ldquoA theorem on tait colorings with an applicationto the generalized Petersen graphsrdquo Journal of CombinatorialTheory vol 6 no 2 pp 152ndash164 1969

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al [6] have succeeded in constructing tree on 119899 vertices119899 ge 5 with locating chromatic numbers varying from 3to 119899 except for (119899 minus 1) Then Behtoei and Omoomi [7]have obtained the locating chromatic number of the Knesergraphs Recently Asmiati et al [8] obtained the locatingchromatic number of the generalized Petersen graph 119875(119899 1)for 119899 ge 3 Baskoro and Asmiati [9] have characterized alltrees with locating chromatic number 3 In [10] all treesof order 119899 with locating chromatic number 119899 minus 1 werecharacterized for any integers 119899 and 119905 where 119899 gt 119905 + 3and 2 le 119905 lt 1198992 Asmiati et al in [11] have succeeded indetermining the locating chromatic number of homogeneousamalgamation of stars and their monotonicity properties andin [12] for firecracker graphs Next Wellyyanti et al [13]determined the locating chromatic number for complete 119899-ary trees

The generalized Petersen graph 119875(119899119898) 119899 ge 3 and 1 le119898 le lfloor(119899 minus 1)2rfloor consists of an outer 119899-cycle 1199101 1199102 119910119899a set of 119899 spokes 119910119894119909119894 1 le 119894 le 119899 and 119899 edges 119909119894119909119894+1198981 le 119894 le 119899 with indices taken modulo 119899 The generalizedPetersen graphwas introduced byWatkins in [14] Let us notethat the generalized Petersen graph 119875(119899 1) is a prism definedas Cartesian product of a cycle 119862119899 and a path 1198752

Next theorems give the locating chromatic numbers forcomplete graph 119870119899 and generalized Petersen graph 119875(119899 1)

Theorem3 (see [6]) For 119899 ge 2 the locating chromatic numberof complete graph 119870119899 is 119899

Theorem 4 (see [8]) The locating chromatic number ofgeneralized Petersen graph 119875(119899 1) is 4 for odd 119899 ge 3 or 5 foreven 119899 ge 4

The barbell graph is constructed by connecting twoarbitrary connected graphs119866 and119867 by a bridge In this paperfirstly we discuss the locating chromatic number for barbellgraph 119861119898119899 for 119898 119899 ge 3 where 119866 and 119867 are complete graphson119898 and 119899 vertices respectively Secondly we determine thelocating chromatic number of barbell graph 119861119875(1198991) for 119899 ge 3where 119866 and119867 are two isomorphic copies of the generalizedPetersen graph 119875(119899 1)

2 Results and Discussion

Next theoremproves the exact value of the locating chromaticnumber for barbell graph 119861119899119899

Theorem 5 Let 119861119899119899 be a barbell graph for 119899 ge 3 Then thelocating chromatic number of 119861119899119899 is 120594119871(119861119899119899) = 119899 + 1

Proof Let 119861119899119899 119899 ge 3 be the barbell graph with the vertexset 119881(119861119899119899) = 119906119894 V119894 1 le 119894 le 119899 and the edge set 119864(119861119899119899)= ⋃119899minus1119894=1 119906119894119906119894+119895 1 le 119895 le 119899 minus 119894 cup ⋃119899minus1119894=1 V119894V119894+119895 1 le 119895 le119899 minus 119894 cup 119906119899V119899

First we determine the lower bound of the locatingchromatic number for barbell graph 119861119899119899 for 119899 ge 3 Sincethe barbell graph 119861119899119899 contains two isomorphic copies of acomplete graph 119870119899 then with respect to Theorem 3 we have120594119871(119861119899119899) ge 119899 Next suppose that 119888 is a locating coloring

using 119899 colors It is easy to see that the barbell graph 119861119899119899contains two vertices with the same color codes which is acontradiction Thus we have that 120594119871(119861119899119899) ge 119899 + 1

To show that 119899 + 1 is an upper bound for the locatingchromatic number of barbell graph 119861119899119899 it suffices to provethe existence of an optimal locating coloring 119888 119881(119861119899119899) 997888rarr1 2 119899 + 1 For 119899 ge 3 we construct the function 119888 in thefollowing way

119888 (119906119894) = 119894 1 le 119894 le 119899

119888 (V119894) =

119899 for 119894 = 1

119894 for 2 le 119894 le 119899 minus 1

119899 + 1 otherwise

(1)

By using the coloring 119888 we obtain the color codes of 119881(119861119899119899)as follows

119888Π (119906119894)

=

0 for 119894119905ℎ component 1 le 119894 le 119899

2 for (119899 + 1)119905ℎ component 1 le 119894 le 119899 minus 1

1 otherwise

119888Π (V119894) =

0 for 119894119905ℎ component 2 le 119894 le 119899 minus 1

for 119899119905ℎ component 119894 = 1 and

for (119899 + 1)119905ℎ component 119894 = 119899

3 for 1119904119905 component 1 le 119894 le 119899 minus 1

2 for 1119904119905 component 119894 = 119899

1 otherwise

(2)

Since all vertices in 119881(119861119899119899) have distinct color codes thenthe coloring 119888 is desired locating coloring Thus 120594119871(119861119899119899) =119899 + 1

Corollary 6 For 119899119898 ge 3 and 119898 = 119899 the locating chromaticnumber of barbell graph 119861119898119899 is

120594119871 (119861119898119899) = max 119898 119899 (3)

Next theorem provides the exact value of the locatingchromatic number for barbell graph 119861119875(1198991)

Theorem 7 Let 119861119875(1198991) be a barbell graph for 119899 ge 3 Then thelocating chromatic number of 119861119875(1198991) is


120594119871 (119861119875(1198991)) =

4 for odd 119899

5 for even 119899(4)




119888 (119906119894) =

1 for 119894 = 1

3 for even 119894 119894 ge 2

4 for odd 119894 119894 ge 3

119888 (119906119899+119894) =

2 for 119894 = 1

3 for odd 119894 119894 ge 3

4 for even 119894 119894 ge 2

119888 (119908119894) =

1 for odd 119894 119894 le 119899 minus 2

2 for even 119894 119894 le 119899 minus 1

3 for 119894 = 119899

119888 (119908119899+119894) =

1 for even 119894 119894 le 119899 minus 1

2 for odd 119894 119894 le 119899 minus 2

4 for 119894 = 119899

(5)


119888Π (119906119894)

=

119894 for 2119899119889 component 119894 le 119899 + 12






1 otherwise

119888Π (119906119899+119894)

=

119894 for 1119904119905 component 119894 le 119899 + 12






1 otherwise

119888Π (119908119894)

=







1 otherwise

119888Π (119908119899+119894)

=







1 otherwise(6)



119888 (119906119894) =

1 for 119894 = 1



5 for 119894 = 119899


119888 (119906119899+119894) =

2 for 119894 = 1

3 for odd 119894 119894 ge 3

4 for even 119894 119894 ge 2

119888 (119908119894) =

1 for odd 119894 119894 le 119899 minus 3

2 for even 119894 119894 le 119899 minus 2

3 for 119894 = 119899 minus 1

4 for 119894 = 119899

119888 (119908119899+119894) =

1 for even 119894 119894 le 119899 minus 2

2 for odd 119894 119894 le 119899 minus 1

5 for 119894 = 119899(7)


119888Π (119906119894)

=








2 for 4119905ℎ component 119894 = 1

for 3119905ℎ component 119894 = 119899

1 otherwise

119888Π (119906119899+119894)

=

119894 for 1119904119905 component 119894 le 1198992







2 for 3119905ℎ component 119894 = 1

1 otherwise

119888Π (119908119894)

=










for 2119899119889 component 119894 = 119899

1 otherwise

119888Π (119908119899+119894)

=










1 otherwise(8)


Data Availability




Acknowledgments



References






















Journal of












Journal of







Volume 2018






Volume 2018









120594119871 (119861119875(1198991)) =

4 for odd 119899

5 for even 119899(4)




119888 (119906119894) =

1 for 119894 = 1

3 for even 119894 119894 ge 2

4 for odd 119894 119894 ge 3

119888 (119906119899+119894) =

2 for 119894 = 1

3 for odd 119894 119894 ge 3

4 for even 119894 119894 ge 2

119888 (119908119894) =

1 for odd 119894 119894 le 119899 minus 2

2 for even 119894 119894 le 119899 minus 1

3 for 119894 = 119899

119888 (119908119899+119894) =

1 for even 119894 119894 le 119899 minus 1

2 for odd 119894 119894 le 119899 minus 2

4 for 119894 = 119899

(5)


119888Π (119906119894)

=

119894 for 2119899119889 component 119894 le 119899 + 12






1 otherwise

119888Π (119906119899+119894)

=

119894 for 1119904119905 component 119894 le 119899 + 12






1 otherwise

119888Π (119908119894)

=







1 otherwise

119888Π (119908119899+119894)

=







1 otherwise(6)



119888 (119906119894) =

1 for 119894 = 1



5 for 119894 = 119899


119888 (119906119899+119894) =

2 for 119894 = 1

3 for odd 119894 119894 ge 3

4 for even 119894 119894 ge 2

119888 (119908119894) =

1 for odd 119894 119894 le 119899 minus 3

2 for even 119894 119894 le 119899 minus 2

3 for 119894 = 119899 minus 1

4 for 119894 = 119899

119888 (119908119899+119894) =

1 for even 119894 119894 le 119899 minus 2

2 for odd 119894 119894 le 119899 minus 1

5 for 119894 = 119899(7)


119888Π (119906119894)

=








2 for 4119905ℎ component 119894 = 1

for 3119905ℎ component 119894 = 119899

1 otherwise

119888Π (119906119899+119894)

=

119894 for 1119904119905 component 119894 le 1198992







2 for 3119905ℎ component 119894 = 1

1 otherwise

119888Π (119908119894)

=










for 2119899119889 component 119894 = 119899

1 otherwise

119888Π (119908119899+119894)

=










1 otherwise(8)


Data Availability




Acknowledgments



References






















Journal of












Journal of







Volume 2018






Volume 2018









119888 (119906119899+119894) =

2 for 119894 = 1

3 for odd 119894 119894 ge 3

4 for even 119894 119894 ge 2

119888 (119908119894) =

1 for odd 119894 119894 le 119899 minus 3

2 for even 119894 119894 le 119899 minus 2

3 for 119894 = 119899 minus 1

4 for 119894 = 119899

119888 (119908119899+119894) =

1 for even 119894 119894 le 119899 minus 2

2 for odd 119894 119894 le 119899 minus 1

5 for 119894 = 119899(7)


119888Π (119906119894)

=








2 for 4119905ℎ component 119894 = 1

for 3119905ℎ component 119894 = 119899

1 otherwise

119888Π (119906119899+119894)

=

119894 for 1119904119905 component 119894 le 1198992







2 for 3119905ℎ component 119894 = 1

1 otherwise

119888Π (119908119894)

=










for 2119899119889 component 119894 = 119899

1 otherwise

119888Π (119908119899+119894)

=










1 otherwise(8)


Data Availability




Acknowledgments



References






















Journal of












Journal of







Volume 2018






Volume 2018









References






















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

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