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Open Problems


TitleAuthor(s)Imp.¹Rec.²sort iconArea » Topic » SubtopicPosted by
Simultaneous partition of hypergraphsKühn; Osthus✭✭0Graph Theory » Hypergraphsfhavet
Odd-cycle transversal in triangle-free graphsErdos; Faudree; Pach; Spencer✭✭0Graph Theory » Extremal G.T.fhavet
Triangle-packing vs triangle edge-transversal.Tuza✭✭0Graph Theory » Extremal G.T.fhavet
Acyclic list colouring of planar graphs.Borodin; Fon-Der-Flasss; Kostochka; Raspaud; Sopena✭✭✭0Graph Theory » Coloring » Vertex coloringfhavet
Steinberg's conjecture✭✭✭✭0Graph Theory » Coloring » Vertex coloringfhavet
Every 4-connected toroidal graph has a Hamilton cycleGrunbaum; Nash-Williams✭✭0Graph Theory » Topological G.T.fhavet
Switching reconstruction conjectureStanley✭✭0Graph Theoryfhavet
Switching reconstruction of digraphsBondy; Mercier✭✭0Graph Theoryfhavet
Hamilton cycle in small d-diregular graphsJackson✭✭0Graph Theory » Directed Graphsfhavet
Edge-disjoint Hamilton cycles in highly strongly connected tournaments.Thomassen✭✭0Graph Theory » Directed Graphs » Tournamentsfhavet
Hoàng-Reed ConjectureHoang; Reed✭✭✭0Graph Theory » Directed Graphsfhavet
Every prism over a 3-connected planar graph is hamiltonian.Kaiser; Král; Rosenfeld; Ryjácek; Voss✭✭0Graph Theory » Basic G.T. » Cyclesfhavet
4-connected graphs are not uniquely hamiltonianFleischner✭✭0Graph Theory » Basic G.T. » Cyclesfhavet
Turán's problem for hypergraphsTuran✭✭0Graph Theory » Hypergraphsfhavet
Hamilton decomposition of prisms over 3-connected cubic planar graphsAlspach; Rosenfeld✭✭0Graph Theory » Basic G.T. » Cyclesfhavet
List chromatic number and maximum degree of bipartite graphsAlon✭✭0Graph Theory » Coloring » Vertex coloringfhavet
Colouring the square of a planar graphWegner✭✭0Graph Theory » Coloring » Vertex coloringfhavet
Weighted colouring of hexagonal graphs.McDiarmid; Reed✭✭0Graph Theory » Coloring » Vertex coloringfhavet
Partitionning a tournament into k-strongly connected subtournaments.Thomassen✭✭0Graph Theory » Directed Graphs » Tournamentsfhavet
PTAS for feedback arc set in tournamentsAilon; Alon✭✭0Graph Theory » Graph Algorithmsfhavet
Decomposing k-arc-strong tournament into k spanning strong digraphsBang-Jensen; Yeo✭✭0Graph Theory » Directed Graphs » Tournamentsfhavet
The Bollobás-Eldridge-Catlin Conjecture on graph packing✭✭✭0Graph Theory » Extremal G.T.asp
Signing a graph to have small magnitude eigenvaluesBilu; Linial✭✭0Graph Theorymdevos
Are almost all graphs determined by their spectrum?✭✭✭0Graph Theorymdevos
Closing Lemma for Diffeomorphism (Dynamical Systems)Charles Pugh✭✭✭✭0TopologyJailton Viana
Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems)✭✭✭✭0TopologyJailton Viana
Arc-disjoint directed cycles in regular directed graphsAlon; McDiarmid; Molloy✭✭0Graph Theory » Directed Graphsfhavet
Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournamentYuster✭✭0Graph Theoryfhavet
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What is the largest graph of positive curvature?DeVos; Mohar✭1Graph Theory » Topological G.T. » Planar graphsmdevos
Bounded colorings for planar graphsAlon; Ding; Oporowski; Vertigan✭✭1Graph Theory » Topological G.T. » Coloringmdevos
A conjecture on iterated circumcentresGoddyn✭✭1Geometrymdevos
Linear-size circuits for stable $0,1 < 2$ sorting?Regan✭✭1Theoretical Comp. Sci. » ComplexityKWRegan
Geodesic cycles and Tutte's TheoremGeorgakopoulos; Sprüssel✭✭1Graph Theory » Basic G.T. » CyclesAgelos
Does every odd number coprime to its Euler totient divide some Carmichael number?Michon✭✭1Number Theory » Computational N.T.maxal
Divisibility of central binomial coefficientsGraham✭✭1Number Theory » Combinatorial N.T.maxal
A sextic counterexample to Euler's sum of powers conjectureEuler✭✭1Number Theory » Computational N.T.maxal
Counterexamples to the Baillie-PSW primality test✭✭1Number Theory » Computational N.T.maxal
Coloring and immersionAbu-Khzam; Langston✭✭✭1Graph Theory » Coloring » Vertex coloringmdevos
Big Line or Big Clique in Planar Point SetsKara; Por; Wood✭✭1GeometryDavid Wood
Matchings extend to Hamiltonian cycles in hypercubesRuskey; Savage✭✭1Graph Theory » Basic G.T. » MatchingsJirka
Seymour's Second Neighbourhood ConjectureSeymour✭✭✭1Graph Theory » Directed Graphsnkorppi
Snevily's conjectureSnevily✭✭✭1Number Theory » Combinatorial N.T.mdevos
MacEachen ConjectureMcEachen✭1Number Theorybillymac00

Imp.¹: Importance (Low ✭, Medium ✭✭, High ✭✭✭, Outstanding ✭✭✭✭)
Rec.²: Recommended for undergraduates.

Note: Resolved problems from this section may be found in Solved problems.

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