1. A Survey on the 3-Decomposition Conjecture Arthur Hoffmann-Ostenhof, Technical University of Vienna Pilsen, 2015

2. 3-Decomposition Conjecture (3DC), AHO, 2009 Every connected cubic graph has a decompositon into a spanning tree, a set of cycles and a matching. Theorem (C.Thomassen, B.Toft, 1979) Every connected cubic graph G has a cycle C such that G-EC is connected.

3. Question (Malkevitch, 1979) Which cubic graphs have a decompositon into a spanning tree, and a set of cycles? Open Problem (Albertson, Berman, Hutchingson, Thomassen, 1990) Is there for each k > 1 a cyclically k-edge connected cubic graph without such a decomposition? History

4. Special Cases Theorem 1 (AHO) Every hamiltonian cubic graph has a 3D. Definition 1 A 3-decomposition (3D) of a cubic graph G is a decompositon of G into a spanning tree, a family of cycles and a matching. Proof:

5. Theorem 2 (K.Ozeki, D.Ye, 2014) Every 3-connected cubic plane graph has a 3D. Every 3-connected cubic graph embedded on the projective plane has a 3D. Theorem 3 (A. Bachstein, D. Ye, 2015) Every 3-connected cubic graph embedded on the torus has a 3D. Every 3-connected cubic graph embedded on the Klein bottle has a 3D. Best Results

6. Theorem 4 (AHO) The Strong-3DC and the 3DC are equivalent. Variations of the 3DC Strong-3DC Let G be a connected cubic graph and C be a 2-regular subgraph of G such that G-EC is connected. Then there is a 3D of G such that the set of cycles of the 3D contains every cycle of C. 3-Decomposition Conjecture (3DC) Every connected cubic graph has a 3D, i.e. a decompositon into a spanning tree, a set of cycles and a matching.

7. Variations of the 3DC 2-Decomposition Conjecture (2DC) Let G be a connected graph with vertices only of degree 2 and 3 such that for every cycle C of G, G-EC is not connected. Then there is a decomposition of G into a spanning tree and a matching. Theorem 5 (AHO) The 3DC, the and the 2DC are equivalent. Theorem (Y.Wang and Q. Zhang,) (nur mündlich?) Every planar graph with girth at least 8 has a 2D. Proof:

8. Related Results (2DC) 2-Decomposition Conjecture (2DC) Let G be a connected graph with vertices only of degree 2 and 3 such that for every cycle C of G, G-EC is not connected. Then there is a decomposition of G into a spanning tree and a matching. Theorem 6 (Y.Wang and Q. Zhang, 2011) Every planar graph with girth at least 8 has a 2D.

9. Definiton 2 A tree T of a plane graph G is called free, if for every vertex v of G, the edges of T incident with v are not in the same facial cycle of G. Free-Tree Conjecture (AHO) Let G be a connected plane graph where every facial cycle has length 2 or 3 and where every vertex of G is contained in a cycle of length 2. Then G has a spanning free tree. 3DC - Planar Variations Theorem 8 (AHO) The Free-Tree Conjecture, the planar-3DC and the planer-2DC are equivalent.

10. THANKS FOR YOUR ATTENTION

11. Theorem (Albertson, Berman,Hutchingson, Thomassen; 1990) Every connected graph G with n vertices has a HIST if δ(G) is at least 4(2n)^1/2. Generalized Question for graphs with higher vertex degree Which graphs have a HIST? (Homeomorphically Irreducible Spanning Tree) Theorem (Lemke, 1988) It is NP-complete to determine whether a given cubic graph has a decomposition into a spanning tree and cycles..