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Decomposing k-arc-strong tournament into k spanning strong digraphs ★★

Author(s): Bang-Jensen; Yeo

Conjecture   Every k-arc-strong tournament decomposes into k spanning strong digraphs.

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PTAS for feedback arc set in tournaments ★★

Author(s): Ailon; Alon

Question   Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments?

Keywords: feedback arc set; PTAS; tournament

Partitionning a tournament into k-strongly connected subtournaments. ★★

Author(s): Thomassen

Problem   Let $ k_1, \dots , k_p $ be positve integer Does there exists an integer $ g(k_1, \dots , k_p) $ such that every $ g(k_1, \dots , k_p) $-strong tournament $ T $ admits a partition $ (V_1\dots , V_p) $ of its vertex set such that the subtournament induced by $ V_i $ is a non-trivial $ k_i $-strong for all $ 1\leq i\leq p $.

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Weighted colouring of hexagonal graphs. ★★

Author(s): McDiarmid; Reed

Conjecture   There is an absolute constant $ c $ such that for every hexagonal graph $ G $ and vertex weighting $ p:V(G)\rightarrow \mathbb{N} $, $$\chi(G,p) \leq \frac{9}{8}\omega(G,p) + c $$

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Colouring the square of a planar graph ★★

Author(s): Wegner

Conjecture   Let $ G $ be a planar graph of maximum degree $ \Delta $. The chromatic number of its square is
    \item at most $ 7 $ if $ \Delta =3 $, \item at most $ \Delta+5 $ if $ 4\leq\Delta\leq 7 $, \item at most $ \left\lfloor\frac32\,\Delta\right\rfloor+1 $ if $ \Delta\ge8 $.

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List chromatic number and maximum degree of bipartite graphs ★★

Author(s): Alon

Conjecture   There is a constant $ c $ such that the list chromatic number of any bipartite graph $ G $ of maximum degree $ \Delta $ is at most $ c \log \Delta $.

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Hamilton decomposition of prisms over 3-connected cubic planar graphs ★★

Author(s): Alspach; Rosenfeld

Conjecture   Every prism over a $ 3 $-connected cubic planar graph can be decomposed into two Hamilton cycles.

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Turán's problem for hypergraphs ★★

Author(s): Turan

Conjecture   Every simple $ 3 $-uniform hypergraph on $ 3n $ vertices which contains no complete $ 3 $-uniform hypergraph on four vertices has at most $ \frac12 n^2(5n-3) $ hyperedges.
Conjecture   Every simple $ 3 $-uniform hypergraph on $ 2n $ vertices which contains no complete $ 3 $-uniform hypergraph on five vertices has at most $ n^2(n-1) $ hyperedges.

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4-connected graphs are not uniquely hamiltonian ★★

Author(s): Fleischner

Conjecture   Every $ 4 $-connected graph with a Hamilton cycle has a second Hamilton cycle.

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Every prism over a 3-connected planar graph is hamiltonian. ★★

Author(s): Kaiser; Král; Rosenfeld; Ryjácek; Voss

Conjecture   If $ G $ is a $ 3 $-connected planar graph, then $ G\square K_2 $ has a Hamilton cycle.

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Hoàng-Reed Conjecture ★★★

Author(s): Hoang; Reed

Conjecture   Every digraph in which each vertex has outdegree at least $ k $ contains $ k $ directed cycles $ C_1, \ldots, C_k $ such that $ C_j $ meets $ \cup_{i=1}^{j-1}C_i $ in at most one vertex, $ 2 \leq j \leq k $.

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Edge-disjoint Hamilton cycles in highly strongly connected tournaments. ★★

Author(s): Thomassen

Conjecture   For every $ k\geq 2 $, there is an integer $ f(k) $ so that every strongly $ f(k) $-connected tournament has $ k $ edge-disjoint Hamilton cycles.

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Hamilton cycle in small d-diregular graphs ★★

Author(s): Jackson

An directed graph is $ k $-diregular if every vertex has indegree and outdegree at least $ k $.

Conjecture   For $ d >2 $, every $ d $-diregular oriented graph on at most $ 4d+1 $ vertices has a Hamilton cycle.

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Switching reconstruction of digraphs ★★

Author(s): Bondy; Mercier

Question   Are there any switching-nonreconstructible digraphs on twelve or more vertices?

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Switching reconstruction conjecture ★★

Author(s): Stanley

Conjecture   Every simple graph on five or more vertices is switching-reconstructible.

Keywords: reconstruction

Every 4-connected toroidal graph has a Hamilton cycle ★★

Author(s): Grunbaum; Nash-Williams

Conjecture   Every 4-connected toroidal graph has a Hamilton cycle.

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Acyclic list colouring of planar graphs. ★★★

Author(s): Borodin; Fon-Der-Flasss; Kostochka; Raspaud; Sopena

Conjecture   Every planar graph is acyclically 5-choosable.

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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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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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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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