Post on 19-Dec-2015
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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 .
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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.
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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 1.2 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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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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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
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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.
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n)22(
2/)1( n
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Example : |E(G)|= and then G is 5 edge-
colorable.
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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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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.
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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