1 A survey of ascending subgraph decomposition 胡維新

2 Abstract A graph G with edges is said to have an ascending subgraph decomposition if its edge set can be decomposed into n sets E 1, E 2, …, E n such that for i=1, 2, …, n and each E i induces a subgraph G i such that G i is isomorphic to a subgraph of G i+1 for i=1, 2, …, n-1. Here we will introduce some results of the ASD conjecture.

3 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 E 1, E 2, …, E n which induce G 1, G 2, …, G n such that |E(G i )| < |E(G i+1 )| and G i is isomorphic to a subgraph of G i+1 (denoted by G i ≤ G i+1 ) for i=1, 2, …, n-1. G 1, G 2, …, G n are the members of the ASD. Usually, we let |E(G i )|=i for i=1, 2, …, n-1 and |E(G n )|=n+t, hence only the case when |E(G)|= is considered except for some special class of graph.

4 G1G1 G2G2 G3G3 G4G4 G5G5 Example: 15=

5 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

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

7 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 S n (n edges) then delete S n and G’ by induction.... G’ SnSn

8 Case 2 Δ( G) =n+1 : G=G’union S n+1 (n+1 edges) then delete the star and union by induction. Let the member G i containing the red edge receive an edge of the S n+1 to form a star then we have an ASD with each member a star.... G’ S n+1

9 Example

10 Theorem 1.3 Let G be a graph on edges and |V(G)|=n+3 then G has an ASD with each member a T i for i=1, 2, …, n.(T i 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 T i.... … … T1T1 T2T2 TnTn

11 S n-1... G’ Case 1 Δ(G) =n-1

12 SnSn G’... Case 2 Δ(G) =n

13 Case 3 Δ(G) =n+1 (assume S n+ 1 G’... is in )

14 Case 4 Δ(G) =n+2 Then similar to Case 1, 2, 3 G\T n-k+1 can be decomposed into G n, G n-1, …, G 1 except G n-k+1

15 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) M 1, M 2, …,M k where |M 1 |=|M 2 |= … =|M k | Step 2 : Split M i into G i and G n+1-i for i=1, 2, …, n/2 when n is even. Split M i into G i and G n-i for i=1, 2, …, (n-1)/2 when n is odd.

16 Example : |E(G)|= and then G is 5 edge-colorable. G 10 G8G8 G9G9 G6G6 G7G7 G4G4 G5G5 G2G2 G3G3 G1G1 11=1+1011=2+911=3+811=4+711=5+6 G i =a matching of size i for i=1, 2, …, 10

17 Theorem 3.1 Any split graph on edges has an ASD. v Complete graph Null 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.

18 Example : |E(G)|=

19 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 G i would be linear forest.

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21 Theorem 3.3 Any forest on on with each member a star forest. edges has an ASD

22 Case 1 exists small branches with at least n edges Example : n=10

23 Case 2 exists a big star with more than edges 1

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

25 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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