Graph Theory » Basic G.T. » Paths

Partition of a cubic 3-connected graphs into paths of length 2. ★★

Author(s): Kelmans

Problem   Does every $ 3 $-connected cubic graph on $ 3k $ vertices admit a partition into $ k $ paths of length $ 2 $?

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Posted by fhavet
updated March 4th, 2013
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Graph Theory » Basic G.T. » Cycles

Decomposing an eulerian graph into cycles with no two consecutives edges on a prescribed eulerian tour. ★★

Author(s): Sabidussi

Conjecture   Let $ G $ be an eulerian graph of minimum degree $ 4 $, and let $ W $ be an eulerian tour of $ G $. Then $ G $ admits a decomposition into cycles none of which contains two consecutive edges of $ W $.

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Posted by fhavet
updated March 4th, 2013
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Graph Theory » Basic G.T. » Cycles

Decomposing an eulerian graph into cycles. ★★

Author(s): Hajós

Conjecture   Every simple eulerian graph on $ n $ vertices can be decomposed into at most $ \frac{1}{2}(n-1) $ cycles.

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Posted by fhavet
updated March 4th, 2013
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Graph Theory » Basic G.T. » Paths

Decomposing a connected graph into paths. ★★★

Author(s): Gallai

Conjecture   Every simple connected graph on $ n $ vertices can be decomposed into at most $ \frac{1}{2}(n+1) $ paths.

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Posted by fhavet
updated March 4th, 2013
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Graph Theory » Coloring » Vertex coloring

Melnikov's valency-variety problem ★

Author(s): Melnikov

Problem   The valency-variety $ w(G) $ of a graph $ G $ is the number of different degrees in $ G $. Is the chromatic number of any graph $ G $ with at least two vertices greater than $$\ceil{ \frac{\floor{w(G)/2}}{|V(G)| - w(G)} } ~ ?$$

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Posted by asp
updated March 3rd, 2013
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Graph Theory

Do any three longest paths in a connected graph have a vertex in common? ★★

Author(s): Gallai

Conjecture   Do any three longest paths in a connected graph have a vertex in common?

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Posted by fhavet
updated October 14th, 2023
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