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 iti.zcu.cz/plzen15/talks/1-2a-Arthur-Survey_decomposition.ppt (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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Bibliography

[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): www.rwoodroofe.math.msstate.edu/workshop2014/bachstein_slides.pdf\\ [5] Arthur Hoffmann-Ostenhof (talk): www.iti.zcu.cz/plzen15/talks/1-2a-Arthur-Survey_decomposition.ppt\\ [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.


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