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黃國卿靜宜大學應用數學系

2

• Let G be an undirected simple graph and H be a subgraph of G.

• G is H-decomposable, denoted by H|G, if its edge set E(G) can be decomposed into subgraphs such that each subgraph is isomorphic to H.

• G has an H-decomposition if G is H-decomposable.

DefinitionsDefinitionsDefinitionsDefinitions

3

2,2,2K

2,2,23 | KS2,2,23 | KK

2,2,24 | KP2,2,23 | KM

2,2,223 | KPP

4

4,1,14 | KP 4,1,1K

H

Hbut is not -decomposable

for other of size 3.

4,1,1K

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MotivationMotivation MotivationMotivation

(Chartrand, Saba and Mynhardt;1994)

4p

4P),3(mod0q

GIfIf is a 2-connected graph of orderis a 2-connected graph of order

sizesize thenthen isis -decomposable.-decomposable.G

andand

Conjecture 1.Conjecture 1.Conjecture 1.Conjecture 1.

,2G

H

IfIf is a graph of sizeis a graph of size andand

thenthen isis -decomposable for some graphdecomposable for some graph

size 3.size 3.

)3(mod0qG

G H ofof

(Chartrand, Saba and Mynhardt;1994)Conjecture 2.Conjecture 2.Conjecture 2.Conjecture 2.

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• It is interesting to us for studying the H-decompositions of a graph G with H of size at most three.

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.1)(1 20 KHHq

It is trivial that for any graph with It is trivial that for any graph with

at least one edge. at least one edge.

GK |2G

8

20 2)(2 MHHq oror .3P

(Chartrand, Polimeni and Stewart.)

Every nontrivial connected graph of even size Every nontrivial connected graph of even size

is -decomposable. is -decomposable. 3P

Theorem 1.Theorem 1.Theorem 1.Theorem 1.

(Chen and Huang)Theorem 2.Theorem 2.Theorem 2.Theorem 2.

Suppose G is a graph of even size and different

from K3∪K2 . Then G is M2-decomposable if and

only if q(G) ≧2Δ(G).

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• The Conjecture 1 is not true in general.

oror 323,1430 ,,,3)(3 PPKPKHHq .3M

CounterexamplesCounterexamples CounterexamplesCounterexamples

(C. Sunil Kumar)(C. Sunil Kumar)

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)3(mod02,| ,,,,4 2121

rr nnnnnn KqrKP

3K .3,1K

except orexcept or

Theorem 3.Theorem 3.Theorem 3.Theorem 3. (C. Sunil Kumar)(C. Sunil Kumar)

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1. -packings of graphs .-packings of graphs .1. -packings of graphs .-packings of graphs . 23 PP

Main resultsMain results Main resultsMain results

2. H-decompositions of graphs with H of size three.-decompositions of graphs with H of size three.2. H-decompositions of graphs with H of size three.-decompositions of graphs with H of size three.

3. -decomposability of graphs .3. -decomposability of graphs .kM

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H -packing.-packing.

)()()()()(21

LEGEGEGEGErnnn

)(),(,),(),(21

LEGEGEGErnnn

HGin ri ,,2,1 for for

L : leave : leave

If thenIf then,L .| GH

are mutually disjoint.

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)( 23 PP -packing-packing

3: PL 2: ML

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SupposeSuppose is a graph different from withis a graph different from with

and Then has and Then has

a -packing with leave a -packing with leave LL where where

13,1,1 cKG

6)(,5)( GqGp .2)( G G

)( 23 PP

).3(mod2)(

);3(mod1)(

);3(mod0)(

3

2

GqifP

GqifP

Gqif

L

Theorem 4.Theorem 4.Theorem 4.Theorem 4.

1. -packings of graphs .-packings of graphs .1. -packings of graphs .-packings of graphs . 23 PP

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(1) and (2-regular) (1) and (2-regular) 2)( G .2)( G

(2) and(2) and2)( G .3)( G

(3) and (3-regular) (3) and (3-regular) 3)( G .3)( G

(4) and(4) and3)( G .4)( G

Proof.Proof.Proof.Proof. Induction onInduction on ).(Gp

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,2G

H

IfIf is a graph of sizeis a graph of size andand

thenthen isis -decomposable for some graphdecomposable for some graph

size 3.size 3.

)3(mod0qG

G H ofof

Conjecture 2.Conjecture 2.Conjecture 2.Conjecture 2.

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Theorem 5.Theorem 5.Theorem 5.Theorem 5.

The Conjecture 2 is affirmative.

Proof.Proof.Proof.Proof.

(1) If q(G) = 3, then G|G.

(2) If G = K4 or K1,1,3c+1, then P4|G.

(3) Otherwise, by Theorem 4, we have (P3∪P2)|G.

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2.1 HH-decompositions of complete multipartite graphs.-decompositions of complete multipartite graphs.

2.2 HH-decompositions of cubic graphs.-decompositions of cubic graphs.

2.3 HH-decompositions of hypercubes.-decompositions of hypercubes.

2. H-decompositions of graphs with H of size three.-decompositions of graphs with H of size three.2. H-decompositions of graphs with H of size three.-decompositions of graphs with H of size three.

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2.1 H-decompositions of complete multipartite graphs. 2.1 H-decompositions of complete multipartite graphs. 2.1 H-decompositions of complete multipartite graphs. 2.1 H-decompositions of complete multipartite graphs.

.13,1,1 cK

)3(mod0| ,,,,3 2121

rr nnnnnn KqKS

Gandand oror4K

1111

)3(mod0| ,,,,32 2121

rr nnnnnn KqKPP

3,1Kandand G .13,1,1 cKoror,4K

2222

13,1,1 cK,3,1K

)3(mod0| ,,,,3 2121

rr nnnnnn KqKM

G ,3,2K ,3,3,1K

.,1,1,1 mK

andand

oror

3333

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• The K3-decomposability of complete multipartite graphs is still widely open.

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2.2 H-decompositions of cubic graphs.2.2 H-decompositions of cubic graphs.2.2 H-decompositions of cubic graphs.2.2 H-decompositions of cubic graphs.

A cubic graph is a 3-regular graph. Let be A cubic graph is a 3-regular graph. Let be

a cubic graph. By the a cubic graph. By the degree-sum formuladegree-sum formula, we , we

obtain Hence, obtain Hence,

G

.2

)(3)(

GpGq ).3(mod0)( Gq

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Suppose Suppose is a cubic graph.is a cubic graph. G

(1)(1) is not is not -decomposable.-decomposable. 3KG

4P(2)(2) isis -decomposable if-decomposable if G G is 2-connected. is 2-connected.

3S(3)(3) isis -decomposable if and only if it is bipartite. -decomposable if and only if it is bipartite. G

.4KG 3M(5)(5) isis -decomposable except -decomposable except G

23 PP (4)(4) isisG .4KG -decomposable except -decomposable except

Theorem 6.Theorem 6.Theorem 6.Theorem 6.

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4p

4P),3(mod0q

GIfIf is a 2-connected graph of orderis a 2-connected graph of order

sizesize thenthen isis -decomposable.-decomposable.G

andand

Conjecture 1.Conjecture 1.Conjecture 1.Conjecture 1.

(Chen and Huang)

3G

),3(mod0q

IfIf is 2-connected ,is 2-connected , and sizeand size

then then

G

4P isis -decomposable.-decomposable.G

Conjecture 3.Conjecture 3.Conjecture 3.Conjecture 3.

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2.3 H-decompositions of hypercubes.2.3 H-decompositions of hypercubes.2.3 H-decompositions of hypercubes.2.3 H-decompositions of hypercubes.

(1) An -cube, denoted by(1) An -cube, denoted byn nQ

.2n21 KQ 21 KQQ nn and forand for

).3(mod0)3(mod02)( 1 nnQq nn(2)(2)

(3) is bipartite is not -decomposable.(3) is bipartite is not -decomposable.nQ nQ 3K

21 KQ 212 KQQ 223 KQQ

, is defined recursively by

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(4) for(4) for3| QH .,,, 32334 PPMSPH

3Q 3Q

234 KQQ

3Q 3Q

3Q 3Q

245 KQQ

3Q 3Q

3Q 3Q

3Q 3Q

3Q 3Q

256 KQQ

(5)

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3Q 3Q3Q 3Q3Q 3Q

3Q 3Q

3Q 3Q

3Q 3Q

3Q 3Q6Q 6Q

6Q 6Q

6Q 6Q

6Q 6Q

9Q

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Suppose and is a graph of Suppose and is a graph of

size 3. Then is -decomposable if size 3. Then is -decomposable if

is different fromis different from

)3(mod0n H

.3K

nQ H H

Theorem 7.Theorem 7.Theorem 7.Theorem 7.

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3. -decomposability of graphs .3. -decomposability of graphs .kM

.| GM k0d(1) If then(1) If then

kMG0d(2) If then is not -decomposable.(2) If then is not -decomposable.

.' GG GM k |.0d(3) Suppose Then(3) Suppose Then

,GkEd LetLet 1k ).(mod0 kE andandTheorem 8.Theorem 8.Theorem 8.Theorem 8.

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Theorem 9.Theorem 9.Theorem 9.Theorem 9.

For a simple graphFor a simple graph .1)()(')(, GGGG

Theorem 10.Theorem 10.Theorem 10.Theorem 10.

G Suppose is a simple graph Suppose is a simple graph and and

Then is equitably -edge colorable. Then is equitably -edge colorable.

G ).(' Gn

G n

(Vizing)

(De Werra)

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.|3 GM0d(1) If then(1) If then

3MG0d(2) If then is not -decomposable.(2) If then is not -decomposable.

.' GG GM |3.0d(3) Suppose Then(3) Suppose Then

GEd 3LetLet ).3(mod0EandandCorollary 11.Corollary 11.Corollary 11.Corollary 11.

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.321 rnnnn W.l.o.g. , assumeW.l.o.g. , assume

0d (1)

0d (2)

3,3,113,1,13,23,1 ,,, KKKKG c .,1,1,1 mKoror

,,,,,, 2,2,2,14,2,1,14,4,16,3,12,2,2,3 KKKKKKG m

3,1,1,1,1K .61,1,1,1,1,1 KK oror

Proof ofProof of . 33

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Conjecture 4.Conjecture 4.Conjecture 4.Conjecture 4.

If If and thenand then

)()( GkGq ,12)( kG ).()(' GG

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Theorem 12.Theorem 12.Theorem 12.Theorem 12.

Theorem 13.Theorem 13.Theorem 13.Theorem 13.

Suppose G is a graph of size q(G) = 2Δ(G). Then χ’(G) = Δ(G) + 1 if and only if G = K3∪K2.

Suppose G is a graph of size q(G) = 3Δ(G) and Δ(G) ≧5. Then χ’(G) = Δ(G).

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RemarkRemarkRemarkRemark

,1kFor there is a graph such that For there is a graph such that

andand

),()( GkGq

12)( kG .1)()(' GG

G

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Proof.Proof.Proof.Proof. Let Let .2 ik

Case1. Then where Case1. Then where .)1(2,,4,2 ki 1)1(2 nik PKG

.2

)2)(1( ikin

.12,,5,3 ki 1)2(2 )\\( nik PeMKG

),12(2

1)2( 2 iikin M

)2(2 ikK

Case2. Then Case2. Then

where is maximum where is maximum

matching of andmatching of and ., Mxxye

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ExampleExampleExampleExample

Case1. Case1. .6,4,2i

.7)('6 47 GPKG

.5)('4 75 GPKG

.3)('2 63 GPKG

Case2. Case2. .7,5,3i

.6)(')\\(5 47 GPeMKG .4)(')\\(3 65 GPeMKG .2)(')\\(1 43 GPeMKG

.7124 kkFor For

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結論結論 .. 結論結論 ..

Conjecture 00.Conjecture 00.Conjecture 00.Conjecture 00. (C.Sunil Kumar)

G

),3(mod0qG

4P

If If is 3-connected and size is 3-connected and size

then is -decomposable.then is -decomposable.

Theorem 00.Theorem 00.Theorem 00.Theorem 00.

GG

4P

If If is a 2-connected cubic graph, then is a 2-connected cubic graph, then

is -decomposable. is -decomposable.

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Conjecture 00.Conjecture 00.Conjecture 00.Conjecture 00. (Chen, Huang and Tsai)

G),3(mod0q

G

4P

If If is 2-connectedand, and sizeis 2-connectedand, and size

then is -decomposable.then is -decomposable.

3G

Remark 00.Remark 00.Remark 00.Remark 00.

Coniecture4 Coniecture3Coniecture4 Coniecture3

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G (3) is 3-regular.(3) is 3-regular.

(a) connected.(a) connected.

(b) disconnected. (b) disconnected.

Proof.Proof.Proof.Proof.

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3,3K 32 KK

4KG No!!No!!

6)( GpBasic step :Basic step :

(a) connected.(a) connected.

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2211 ,,\ yxyxyxGH Let Let

x y

1y

2y2x

1x

x y

1y

2y2x

1x

8)( GpInduction step :Induction step :

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3222111 ,:1 PPyxvyxcase

323212232111 ,,:2 PPuuuyxvvvyxcase

323212232111 ,,:3 PPuuuyxvvvyxcase

323212232111 ,,:4 PPuuuyxvvvyxcase

x y

1y

2y2x

1x

x y

1y

2y2x

1x

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(b) disconnected(b) disconnected

,214 nGGGmKG

wherewhere 0m andand 4KGi

.1)( 0 ma LetLet .\ 4KGH

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1)( 0 mb

evenmi )(

nGGGKm

G 21422

oddmii )(

nGGGKKm

G

2144 322

3

46

210 t

320 t

tGPP ,|32 2 or 32 or 3