A survey of ascending subgraph
decomposition
胡維新
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Abstract
A graph G with
 n  1

 2 



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 .
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In 1987, Paul Erdös and the others posed the following conjecture.
Ascending Subgraph Decomposition Conjecture :
 n  1

  t
 2 
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(Gi)|=i for i=1, 2, …, n-1 and |E(Gn)|=n+t, hence only the case
 n  1
when |E(G)|=  2  is considered except for some special class of
graph.
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 5  1
Example: 15= 
 2 



G1
G2
G3
G4
G5
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Three directions in dealing with the ASD problem
(1) |V(G)|≤n+3
(2) (G)  (2  2 )n
(3) Special classes of graphs : split graphs,
complete t-partite graphs, forests, regular
graphs
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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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 n  1

 edges
 2 
Theorem 1.2 Let G be a graph on
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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Case 2 Δ(G) =n+1 : G=G’union Sn+1(n+1 edges)
then delete the star and union by induction.
. . .
. . .
G’
Sn+1
Let the member Gi containing the red edge receive an edge of
the Sn+1 to form a star then we have an ASD with each member a
star.
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Example
9
 n  1

 2 
edges


Theorem 1.3 Let G be a graph on
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
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Case 1 Δ(G) =n-1
G’
. . .
Sn-1
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Case 2 Δ(G) =n
G’
. . .
Sn
 (i) 

S n  Gn' 1  Gn  Gn 1  (ii ) 
(iii ) 

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Case 3 Δ(G) =n+1 (assume
'
is in G l )
G’
. . .
Sn+ 1
 (i) 

'
S n 1  Gl  Gn  Gl  (ii ) 
(iii ) 

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Case 4 Δ(G) =n+2
(i)  x s.t. n  1  deg( x)  n  1 then go back to Case 1, 2, 3
(ii ) (i) failed and  x s.t. deg( x)  n  k
Then similar to Case 1, 2, 3 G\Tn-k+1 can be decomposed into Gn, Gn-1, …,
G1 except Gn-k+1
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Theorem 2.1 If a graph G has
G has an ASD.
 n  1


edges,
 2 
and Δ(G)<
 n  1


Theorem 2.2 If a graph G has  2  edges, and Δ(G) ≤
(2 

2, )then
n
(n  1) / 2,
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|= …
=|Mk|
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.
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Example : |E(G)|=
G10
G5
11=1+10
10  1

 and then
 2 
G9
G8
G4
11=2+9
G3
11=3+8
G is 5 edge-colorable.
G7
G6
G2
11=4+7
G1
11=5+6
Gi=a matching of size i for i=1, 2, …, 10
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Theorem 3.1 Any split graph
 n  1
on  2  edges
has an ASD.
v
Null
graph
Complete
graph
. . .
. . .
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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Example :
 5  1
|E(G)|=  2 


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 n  1
  tedges where t < n, has an ASD.
Theorem 3.2 Any r-regular graph G on 
2


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
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valency r’<v/2 and the members Gi would be linear forest.
. . .
. . .
. . .
. . .
. . .
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Theorem 3.3 Any forest on on  n  1 edges has an ASD
with each member a star forest.
 2 
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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
n

 2
 edges


}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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