Graph Theory » Coloring » Vertex coloring

Earth-Moon Problem ★★

Author(s): Ringel

Problem   What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?

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Posted by fhavet
updated March 6th, 2013
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Graph Theory » Extremal G.T.

Triangle-packing vs triangle edge-transversal. ★★

Author(s): Tuza

Conjecture   If $ G $ has at most $ k $ edge-disjoint triangles, then there is a set of $ 2k $ edges whose deletion destroys every triangle.

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Posted by fhavet
updated March 6th, 2013
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Graph Theory » Extremal G.T.

Odd-cycle transversal in triangle-free graphs ★★

Author(s): Erdos; Faudree; Pach; Spencer

Conjecture   If $ G $ is a simple triangle-free graph, then there is a set of at most $ n^2/25 $ edges whose deletion destroys every odd cycle.

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Posted by fhavet
updated March 6th, 2013
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Graph Theory » Hypergraphs

Simultaneous partition of hypergraphs ★★

Author(s): Kühn; Osthus

Problem   Let $ H_1 $ and $ H_2 $ be two $ r $-uniform hypergraph on the same vertex set $ V $. Does there always exist a partition of $ V $ into $ r $ classes $ V_1, \dots , V_r $ such that for both $ i=1,2 $, at least $ r!m_i/r^r -o(m_i) $ hyperedges of $ H_i $ meet each of the classes $ V_1, \dots , V_r $?

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Posted by fhavet
updated March 6th, 2013
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