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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
5
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.
6
• It is interesting to us for studying the H-decompositions of a graph G with H of size at most three.
7
.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).
9
• The Conjecture 1 is not true in general.
oror 323,1430 ,,,3)(3 PPKPKHHq .3M
CounterexamplesCounterexamples CounterexamplesCounterexamples
(C. Sunil Kumar)(C. Sunil Kumar)
10
)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
12
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
14
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
15
(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
16
,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.
17
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.
18
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.
19
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
20
• The K3-decomposability of complete multipartite graphs is still widely open.
21
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
22
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.
23
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.
24
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
25
(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)
26
3Q 3Q3Q 3Q3Q 3Q
3Q 3Q
3Q 3Q
3Q 3Q
3Q 3Q6Q 6Q
6Q 6Q
6Q 6Q
6Q 6Q
9Q
27
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.
28
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.
29
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)
30
.|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.
31
.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
32
Conjecture 4.Conjecture 4.Conjecture 4.Conjecture 4.
If If and thenand then
)()( GkGq ,12)( kG ).()(' GG
33
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).
34
RemarkRemarkRemarkRemark
,1kFor there is a graph such that For there is a graph such that
andand
),()( GkGq
12)( kG .1)()(' GG
G
35
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
36
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
37
38
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.
39
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
40
G (3) is 3-regular.(3) is 3-regular.
(a) connected.(a) connected.
(b) disconnected. (b) disconnected.
Proof.Proof.Proof.Proof.
41
3,3K 32 KK
4KG No!!No!!
6)( GpBasic step :Basic step :
(a) connected.(a) connected.
42
2211 ,,\ yxyxyxGH Let Let
x y
1y
2y2x
1x
x y
1y
2y2x
1x
8)( GpInduction step :Induction step :
43
3222111 ,:1 PPyxvyxcase
323212232111 ,,:2 PPuuuyxvvvyxcase
323212232111 ,,:3 PPuuuyxvvvyxcase
323212232111 ,,:4 PPuuuyxvvvyxcase
x y
1y
2y2x
1x
x y
1y
2y2x
1x
44
(b) disconnected(b) disconnected
,214 nGGGmKG
wherewhere 0m andand 4KGi
.1)( 0 ma LetLet .\ 4KGH
45
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
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