Balanced Path Decompositions of Crowns and Directed Crowns

Hung-Chih Lee and Shun-Li Hsu

Department of Information Technology

Ling Tung University

Taichung, Taiwan 40852, R.O.C.

Outline Introduction

Previous results

Our results

kP

Introduction H-decomposition of G

Balanced H-decomposition of G

kk PP

path directed and Path

kP

*,, crown directed and Crown lnln CC

H-decomposition of G

Let G and H be graphs (digraphs) . An H-decomposition of G is a partition of the edge (arc) set of G into subsets each of which induces a graph (digraph) isomorphic to H.

If G has an H-decomposition, then we say H decomposes G, denoted by H|G.

kP

Balanced H-decomposition of G

An H-decomposition of G is balanced if each vertex of G belongs to the same number of members in the decomposition.

We write H||G If G admits a balanced H-decomposition.

kP

Path and directed path : a path on k vertices

: a directed path on k vertices

kP

kP

P4

5P

Crown :lnC , },,,,,,,{)( 110110, nnln bbbaaaCV

)}(mod1,,1,,1,,1,0:{)( , nliiijnibaCE jiln

)3,5(3,5 lnC

0a 4a2a 3a1a

0b 1b 2b 3b 4b

Properties of the crown

regularlC ln is ,

nnnnnnln KIIlnKC ,,, offactor 1:)(

pqnpn CC ,, |

Directed crown

e

eCC lnln

of sendvertice theconnecting arcs opposite by two

edgeeach replacingby from obtainedgraph the: ,*,

1a 2a0a

0b 1b 2b*

2,3C

0a 2a1a

0b 1b 2b

2,3C

Previous results

Bermond （ 1975 ）

Hung and Mendelsohn（ 1977 ）

Lee and Lin（ 2009 ）

Main results: Theorem A.

Corollary B.

|)1(2|| , lkkCP lnk

lnk CP ,||

|)1(2|| , nkkKP nnk

Main results: Theorem C.

Corollary D.

lkCP lnk |)1(|| *,

*, || lnk CP

nkKP nnk |)1(|| *,

label of edges (arcs)

)(mod)(mod)(

)(mod)(

)(mod)(

nijnijab

nijba

nijba

ij

ji

ji

Example of edges theLabeling 3,5C

Label 0 : Label 1 : Label 2 :

0a 4a2a 3a1a

0b 1b 2b 3b 4b

Example

2a 3a1a

1b 2b 3b*

2,3C

of arcs theLabeling *2,3C

Label 0 :

Label 1 :

: 0 Label

: 1 Label

Notations

Eaxmple

)}(:{)(

,)}(:{)}(:{)(

with ofsubgraph thedenotes Then

.Let

,

,

GEbabatGE

GVbbGVaatGV

CtG

CG

jititi

itiiti

ln

ln

Notations

Example

)}(:{

)}(:{)(

,)}(:{)}(:{)(

with ofsubgraph thedenotes Then

.Let *,

*,

GAabab

GAbabatGA

GVbbGVaatGV

CtG

CG

srtstr

jititi

itiiti

ln

ln

Notations

))()((

.)()( and )()(with

(digraph)sgraph thedenotes , ,

Then .(digraphs) graphs be ,,,Let

1

11

121

21

m

ii

m

ii

m

ii

m

iim

m

FAFA

FEFEFVFV

FFFFF

FFF

Proof of Theorem A

(Necessity) Cn,l is l-regular and by Lemma 2.5

(Sufficiency) 2(k － 1)|lk ⇒ k － 1|l Case 1. k is even

By Lemma 2.2, it suffices to show 1

21

222

2

3120:Let

kkkkkk babababaW

|)1(2|| , lkkCP lnk

1,|| knk CP

ion.decomposit required theis }1,,1,0:{ nttW

0,1,2,,4,3,2 :label kkk

)4,3,5( ofion decomposit- Balanced 3,54 klnCP

0a 4a2a 3a1a

0b 1b 2b 3b 4b

)even ( ofion decomposit- Balanced , kCP lnk

Base graph

0a 1a

1b 2b

0 12

Proof of Theorem A Case 2. k is odd

2(k － 1)|lk ⇒ 2(k － 1)|l. By Lemma 2.2,

it suffices to show

2

1

2

1

2

3

2

13120:Let kkkkkk ababbabaP

|)1(2|| , lkkCP lnk

)1(2,|| knk CP

0,1,2,,4,3,2 :label kkk

Proof of Theorem A

2

53

2

3

2

33

2

5142032:Let kkkkkk babaababQ

|)1(2|| , lkkCP lnk

ion.decomposit required theis

}1,,1,0:,{ nttQtP

1,,1,,52,42,32 :label kkkkkk

)3,4,5( ofion decomposit- Balanced 4,53 klnCP

0a 4a2a 3a1a

0b 1b 2b 3b 4b

) odd( ofion decomposit- Balanced , kCP lnk

0a 1a

1b 2b 3b

0 21

3

Baes graph

Proof of Theorem B

(Necessity) C*n,l is 2l-regular and by Lemma 2.5

(Sufficiency) Case 1. 2(k － 1)|lk

By Theorem A, there exists a balanced Pk-decomposition of C*

n,l

Replace each edge in Cn,l by two arcs with opposite directions⇒ each Pk in becomes two with opposite directions ⇒ Done.

lkCP lnk |)1(|| *,

sPk '

ofion decomposit- Balanced *3,54 CP

0a 4a2a 3a1a

0b 1b 2b 3b 4b

) |)1(2( ofion decomposit- Balanced *, lkkCP lnk

Base graph

0a 1a

1b 2b

02 1

0a 1a

1b 2b

102

Replace each edge in the crown by two arcs with opposite directions

Proof of Theorem B Case 2.

lkCP lnk |)1(|| *,

odd is |)1(2 and |1 klkklk

lkk |)1(2

2

1

2

1

2

3

2

13120:Let kkkkkk ababbabaP

0,1,2,,4,3,2 :label kkk

Proof of Theorem B

lkCP lnk |)1(|| *,

2

3

2

3

2

1

2

51302:Let kkkkkk babaababQ

0,1,2,,4,3,2 :label kkk

ion.decomposit required theis

}1,,1,0:,{ nttQtP

ofion decomposit- Balanced *

2,33 CP

) |)1(2(

ofion decomposit- Balanced *,

lkk

CP lnk

3a

1b 2b 3b

2a1aBase graph

2a1a

Thank you for your attention!

Previous results Path decomposition

Tarsi（ 1983 ） Truszczyński （ 1985 ） Shyu and Lin（ 2003 ） Meszka and Skupień （ 2006 ）

Balanced path decomposition Bermond （ 1975 ） Hung and Mendelsohn（ 1977 ）

Our object

Find the necessary and sufficient conditions for

kP

lnk CP , ofion decomposit- Balanced

*, ofion decomposit- Balanced lnk CP

Main results Main results

Necessary condition － Counting Method

Sufficient condition － Construction Method

lnk CP , ofion decomposit- Balanced

*, ofion decomposit- Balanced lnk CP

Procedure of the proof Labeling the edges of the crown Find a base graph (path)

k is even k is odd

2(k-1) does not divide lk

Shifting the base graph

lnk CP , ofion decomposit- Balanced

*, ofion decomposit- Balanced lnk CP

lkk |)1(2

Future works Find the necessary and sufficient conditions for

lnk CP , ofion decomposit- Balanced *, ofion decomposit- Balanced lnk CP

),( ofion decomposit- Balanced * rnKPk

),( ofion decomposit- Balanced rnKPk

,

0a 2a 3a1a

0b 1b 2b 3b

24C0a 2a 3a1a

0b 1b 2b 3b

G

9 prisoners problem (P3||K9)

vk KP ofion decomposit- Balanced

),,(design Handcuffed kvH