![](/files/happy5.png)
Hoffmann-Ostenhof
3-Edge-Coloring Conjecture ★★★
Author(s): Arthur; Hoffmann-Ostenhof
Conjecture Suppose
with
is a connected cubic graph admitting a
-edge coloring. Then there is an edge
such that the cubic graph homeomorphic to
has a
-edge coloring.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ |V(G)|>2 $](/files/tex/ac99a71a6e28acc3052e542089238e810d347be6.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ e \in E(G) $](/files/tex/730c5d64c8d749c640adc18eb493c641ff1addc9.png)
![$ G-e $](/files/tex/a9c40841d6043b14ca9501d156e86164ad3f81e5.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
Keywords: 3-edge coloring; 4-flow; removable edge
Cycle Double Covers Containing Predefined 2-Regular Subgraphs ★★★
Author(s): Arthur; Hoffmann-Ostenhof
Conjecture Let
be a
-connected cubic graph and let
be a
-regular subgraph such that
is connected. Then
has a cycle double cover which contains
(i.e all cycles of
).
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ 2 $](/files/tex/5271e36bb1c040e0f14061d89cd97d0c86d4e06f.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ 2 $](/files/tex/5271e36bb1c040e0f14061d89cd97d0c86d4e06f.png)
![$ G-E(S) $](/files/tex/f65af23afc4481cc9a13687d7a6d12108bda7714.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
Keywords:
3-Decomposition Conjecture ★★★
Author(s): Arthur; Hoffmann-Ostenhof
Conjecture (3-Decomposition Conjecture) Every connected cubic graph
has a decomposition into a spanning tree, a family of cycles and a matching.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: cubic graph
Strong 5-cycle double cover conjecture ★★★
Author(s): Arthur; Hoffmann-Ostenhof
Conjecture Let
be a circuit in a bridgeless cubic graph
. Then there is a five cycle double cover of
such that
is a subgraph of one of these five cycles.
![$ C $](/files/tex/05d3558ef52267cc1af40e658352d98883668ee3.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ C $](/files/tex/05d3558ef52267cc1af40e658352d98883668ee3.png)
Keywords: cycle cover
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