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黃國卿靜宜大學應用數學系
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• 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
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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.
13
)( 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