Graph Theory » Extremal G.T.

Complexity of the H-factor problem. ★★

Author(s): Kühn; Osthus

An $ H $-factor in a graph $ G $ is a set of vertex-disjoint copies of $ H $ covering all vertices of $ G $.

Problem  Let $ c $ be a fixed positive real number and $ H $ a fixed graph. Is it NP-hard to determine whether a graph $ G $ on $ n $ vertices and minimum degree $ cn $ contains and $ H $-factor?

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Posted by fhavet
updated March 5th, 2013
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Graph Theory » Basic G.T.

Subgraph of large average degree and large girth. ★★

Author(s): Thomassen

Conjecture   For all positive integers $ g $ and $ k $, there exists an integer $ d $ such that every graph of average degree at least $ d $ contains a subgraph of average degree at least $ k $ and girth greater than $ g $.

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Posted by fhavet
updated March 5th, 2013
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Graph Theory

Turán number of a finite family. ★★

Author(s): Erdos; Simonovits

Given a finite family $ {\cal F} $ of graphs and an integer $ n $, the Turán number $ ex(n,{\cal F}) $ of $ {\cal F} $ is the largest integer $ m $ such that there exists a graph on $ n $ vertices with $ m $ edges which contains no member of $ {\cal F} $ as a subgraph.

Conjecture   For every finite family $ {\cal F} $ of graphs there exists an $ F\in {\cal F} $ such that $ ex(n, F ) = O(ex(n, {\cal F})) $ .

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Posted by fhavet
updated March 5th, 2013
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Graph Theory » Directed Graphs

Subdivision of a transitive tournament in digraphs with large outdegree. ★★

Author(s): Mader

Conjecture   For all $ k $ there is an integer $ f(k) $ such that every digraph of minimum outdegree at least $ f(k) $ contains a subdivision of a transitive tournament of order $ k $.

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

Large induced forest in a planar graph. ★★

Author(s): Abertson; Berman

Conjecture   Every planar graph on $ n $ verices has an induced forest with at least $ n/2 $ vertices.

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Posted by fhavet
updated March 4th, 2013
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Graph Theory

Lovász Path Removal Conjecture ★★

Author(s): Lovasz

Conjecture   There is an integer-valued function $ f(k) $ such that if $ G $ is any $ f(k) $-connected graph and $ x $ and $ y $ are any two vertices of $ G $, then there exists an induced path $ P $ with ends $ x $ and $ y $ such that $ G-V(P) $ is $ k $-connected.

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Posted by fhavet
updated March 4th, 2013
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