Importance: High ✭✭✭
Subject: Graph Theory
Keywords: cubic graph
Recomm. for undergrads: no
Posted by: arthur
on: January 24th, 2017

\begin{conjecture} (3-Decomposition Conjecture) Every connected cubic graph $G$ has a decomposition into a spanning tree, a family of cycles and a matching. \end{conjecture}

We state the conjecture in a more precise manner:\\

Let $G$ be a connected cubic graph. Then $G$ contains a spanning tree $H_1$, a $2$-regular subgraph $H_2$ and a matching $H_3$ (where only $H_3$ and not $H_1$ or $H_2$ may be empty) such that $E(H_1) \cup E(H_2) \cup E(H_3) = E(G)$ and $E(H_i) \cap E(H_j) =\emptyset$ for every $\{i,j\} \subseteq \{1,2,3\}$ with $i\not=j$.

The conjecture holds for all hamiltionian cubic graphs and for all connected planar cubic graphs, see [1] and see also [7].

Every cubic graph G which has a spanning tree T such that every vertex of T has degree three or one (such spanning tree T is called a HIST) obviously satisfies this conjecture. But not every connected cubic graph has a HIST, see [2].

The 3-Decomposition Conjecture has been shown to be equivalent to the following conjecture:

\begin{conjecture}(2-Decomposition Conjecture) Let $G$ be connected graph where every vertex has degree two or three. Suppose that for every cycle $C$ of $G$, $G-E(C)$ is disconnected, then $G$ has a decomposition into a spanning tree $T$ and a matching $M$, i.e $G-M=T$. \end{conjecture}

Note that every cycle $C$ which passes through a vertex of degree two satisfies the condition that G-E(C) is disconnected.\\

Remark: The 3-Decomposition Conjecture has also been shown to hold for other classes of cubic graphs, see for instance [3,4]. A survey on the 3-Decompostion conjecture has been given by the author 2015 in Pilsen (at that time the planar case was still open) see (and press play if you find the play button). Note that there are several papers on the problem whether a planar graph $G$ has a matching $M$ such that $G-M$ is acyclic, see for instance [6].

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[1] Arthur Hoffmann-Ostenhof, Tomáš Kaiser, Kenta Ozeki, \arXiv[Decomposing planar cubic graphs] 1609.05059 [math.CO]\\ [2] Arthur Hoffmann-Ostenhof, Kenta Ozeki, \arXiv[On HISTs in Cubic Graphs] 1507.07689 [math.CO]\\ [3] F. Abdolhosseini, S. Akbari, H. Hashemi, M.S. Moradian, \arXiv[Hoffmann-Ostenhof's conjecture for traceable cubic graphs] 1607.04768[math.CO] \\ [4] Anna Bachstein, Dong Ye (talk):\\ [5] Arthur Hoffmann-Ostenhof (talk):\\ [6] Yingqian Wang, Qijun Zhang, Discrete Mathematics 311 (2011) 844–849, Decomposing a planar graph with girth at least 8 into a forest and a matching\\ [7] Kenta Ozeki, Dong Ye, Decomposing plane cubic graphs, European Journal of Combinatorics 52 (2016) 40-46.

* indicates original appearance(s) of problem.


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