A Survey on the 3-Decomposition Conjecture

Arthur Hoffmann-Ostenhof, Arthur Hoffmann-Ostenhof,

Technical University of ViennaTechnical University of Vienna

Pilsen, 2015

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.

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

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:

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

Theorem 4 (AHO) The Strong-3DC and the 3DC are equivalent.

Variations of the 3DC

Strong-3DCLet 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.

Variations of the 3DC2-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.

Proof:

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.

Definiton 2A 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.

THANKS FOR YOUR ATTENTIONTHANKS FOR YOUR ATTENTION

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. .