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A survey of ascending subgraph decomposition

胡維新

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Abstract

A graph G with edges is said to have an ascending

subgraph decomposition if its edge set can be decomposed into n

sets E1, E2, …, En such that for i=1, 2, …, n and each Ei induces a

subgraph Gi such that Gi is isomorphic to a subgraph of Gi+1 for

i=1, 2, …, n-1. Here we will introduce some results of the ASD

conjecture .

2

1n

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In 1987, Paul Erdös and the others posed the following conjecture.

Ascending Subgraph Decomposition Conjecture :

Let G be a graph on edges where 0≤t≤n then E(G) can be partitioned into n set E1, E2, …, En which induce G1, G2, …, Gn such that |E(Gi)| < |E(Gi+1)| and Gi is isomorphic to a subgraph of Gi+1

(denoted by Gi ≤ Gi+1 ) for i=1, 2, …, n-1.

G1, G2, …, Gn are the members of the ASD. Usually, we let |E(G

i)|=i for i=1, 2, …, n-1 and |E(Gn)|=n+t, hence only the case when |E(G)|= is considered except for some special class of graph.

tn

2

1

2

1n

4G1 G2G3 G4

G5

Example: 15=

2

15

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Three directions in dealing with the ASD problem

(1) |V(G)|≤n+3

(2)

(3) Special classes of graphs : split graphs, complete t-partite graphs, forests, regular graphs

nG )22()(

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Theorem 1.1 The complete graph Kn+1 has an ASD with each member a star (a path or mixed).

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Theorem 1.2 Let G be a graph on edges and |V(G)|=n+2 then G has an ASD with each member a star.

Proof : n ≤Δ(G) ≤ n+1

Case 1 Δ(G) =n : G=G’ union Sn(n edges) then delete Sn and G’ by induction.

. . .

G’Sn

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1n

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Case 2 Δ(G) =n+1 : G=G’union Sn+1(n+1 edges) then delete the star and union by induction.

Let the member Gi containing the red edge receive an edge of t

he Sn+1 to form a star then we have an ASD with each member a star.

. . .

. . .

G’ Sn+1

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Example

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Theorem 1.3 Let G be a graph on edges and |V(G)|=n+3 then G has an ASD with each member a Ti for i=1, 2, …, n.(Ti is a star union a leg)

Proof : Similar to Theorem 1.2 and consider four cases according to Δ(G) =n-1, n, n+1 or n+2 we could have an ASD with each member a Ti.

. . .

…

…T1 T2 Tn

2

1n

11

Sn-1

. . .

G’

Case 1 Δ(G) =n-1

12

Sn

G’

. . .

Case 2 Δ(G) =n

)(

)(

)(

1'

1

iii

ii

i

GGGS nnnn

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Case 3 Δ(G) =n+1 (assume

)(

)(

)('

1

iii

ii

i

GGGS lnln

Sn+ 1

G’

. . .

'lGis in )

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Case 4 Δ(G) =n+2

3,2,11)deg(1..)( Casetobackgothennxntsxi

knxtsxandfailediii )deg(..)()(

Then similar to Case 1, 2, 3 G\Tn-k+1 can be decomposed into Gn, Gn-1, …, G

1 except Gn-k+1

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Theorem 2.1 If a graph G has edges, and Δ(G)< , then G has an ASD.

Theorem 2.2 If a graph G has edges, and Δ(G) ≤ , then G has an ASD with each a member a matching.

Proof :

Step 1 :

Partitioned the edge set of G into k matchings (k=n/2 or (n+1)/2 according to k is even or odd) M1, M2, …,Mk where |M1|=|M2|= … =|M

k|

Step 2 :

Split Mi into Gi and Gn+1-i for i=1, 2, …, n/2 when n is even.

Split Mi into Gi and Gn-i for i=1, 2, …, (n-1)/2 when n is odd.

2

1n

2

1n

n)22(

2/)1( n

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Example : |E(G)|= and then G is 5 edge-

colorable.

2

110

G10 G8G9 G6G7

G4G5 G2G3 G1

11=1+10 11=2+9 11=3+8 11=4+7 11=5+6

Gi=a matching of size i for i=1, 2, …, 10

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Theorem 3.1 Any split graph on edges has an

ASD.

v

Completegraph

Nullgraph

. . .

. . .

Proof : Delete a star of n edges from the edges from the edges incident to v (the edges between null graph and complete graph first) and the by induction.

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1n

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Example : |E(G)|=

2

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Theorem 3.2 Any r-regular graph G on edges where t < n, has an ASD.Proof :Case 1. r ≤ n/2, then by Thm 2.2 with each member a matching.

Case 2. n/2<r ≤2n/3 :

Case 3. 2n/3<r<v/2:

Case 4. r≥v/2. Peel off Hamiltonian cycles from the graph until the remaining valency r’<v/2 and the members Gi would be linear forest.

tn

2

1

20

. . .

. . .

. . .

. . .

. . .

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Theorem 3.3 Any forest on on

2

1n

with each member a star forest.

edges has an ASD

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Case 1 exists small branches with at least n edges

Example : n=10

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Case 2 exists a big star with more than edges

2

n

}1

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Case 3 exists at least two stars with size at least n

} k

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Theorem 3.4 Any complete multipartite graph has an ASD with each member a star or a double star or a pregnant star.

Double star Pregnat star

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