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Open Problems


TitleAuthor(s)Imp.¹Rec.²Area » Topic » SubtopicPosted bysort icon
Graceful Tree Conjecture✭✭✭0Graph Theory » Coloring » Labelingkintali
Twin prime conjecture✭✭✭✭0Number Theorykaushiks.nitt
Euler-Mascheroni constant✭✭✭0Number Theory » Analytic N.T.Juggernaut
The Bermond-Thomassen ConjectureBermond; Thomassen✭✭0Graph Theory » Directed GraphsJS
Mixing Circular ColouringsBrewster; Noel✭1Graph Theory » Coloring » Vertex coloringJon Noel
Choice Number of k-Chromatic Graphs of Bounded OrderNoel✭✭1Graph Theory » Coloring » Vertex coloringJon Noel
Bounding the on-line choice number in terms of the choice numberZhu✭✭1Graph Theory » Coloring » Vertex coloringJon Noel
Choosability of Graph PowersNoel✭✭1Graph Theory » Coloring » Vertex coloringJon Noel
Erdős–Faber–Lovász conjectureErdos; Faber; Lovasz✭✭✭0Graph Theory » Coloring » Vertex coloringJon Noel
2-colouring a graph without a monochromatic maximum cliqueHoang; McDiarmid✭✭0Graph Theory » Coloring » Vertex coloringJon Noel
Kriesell's ConjectureKriesell✭✭0Graph Theory » Basic G.T. » ConnectivityJon Noel
Partitioning the Projective PlaneNoel✭✭1GeometryJon Noel
List Total Colouring ConjectureBorodin; Kostochka; Woodall✭✭0Graph Theory » ColoringJon Noel
List Colourings of Complete Multipartite Graphs with 2 Big PartsAllagan✭✭1Graph Theory » Coloring » Vertex coloringJon Noel
Saturated $k$-Sperner Systems of Minimum SizeMorrison; Noel; Scott✭✭1Combinatorics » PosetsJon Noel
The Double Cap ConjectureKalai✭✭0CombinatoricsJon Noel
Cycles in Graphs of Large Chromatic NumberBrewster; McGuinness; Moore; Noel✭✭0Graph Theory » Coloring » Vertex coloringJon Noel
Saturation in the HypercubeMorrison; Noel; Scott✭✭0CombinatoricsJon Noel
Extremal $4$-Neighbour Bootstrap Percolation in the HypercubeMorrison; Noel✭✭0CombinatoricsJon Noel
Turán Problem for $10$-Cycles in the HypercubeErdos✭✭0CombinatoricsJon Noel
Weak saturation of the cube in the cliqueMorrison; Noel✭1Graph Theory » Extremal G.T.Jon Noel
Sidorenko's ConjectureSidorenko✭✭✭0Graph TheoryJon Noel
Multicolour Erdős--Hajnal ConjectureErdos; Hajnal✭✭✭0Graph Theory » Extremal G.T.Jon Noel
Matchings extend to Hamiltonian cycles in hypercubesRuskey; Savage✭✭1Graph Theory » Basic G.T. » MatchingsJirka
Algorithm for graph homomorphismsFomin; Heggernes; Kratsch✭✭0Graph Theory » Coloring » Homomorphismsjfoniok
Rank vs. GenusJohnson✭✭✭0TopologyJesse Johnson
Approximation Ratio for Maximum Edge Disjoint Paths problemBentz✭✭0Graph Theoryjcmeyer
Approximation ratio for k-outerplanar graphsBentz✭✭0Graph Theoryjcmeyer
Finding k-edge-outerplanar graph embeddingsBentz✭✭0Graph Theoryjcmeyer
Closing Lemma for Diffeomorphism (Dynamical Systems)Charles Pugh✭✭✭✭0TopologyJailton Viana
Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems)✭✭✭✭0TopologyJailton Viana
Partial List ColoringAlbertson; Grossman; Haas✭✭✭0Graph Theory » Coloring » Vertex coloringIradmusa
Partial List ColoringIradmusa✭✭✭0Graph Theory » Coloring » Vertex coloringIradmusa
Total Colouring ConjectureBehzad✭✭✭0Graph Theory » ColoringIradmusa
Vertex Coloring of graph fractional powersIradmusa✭✭✭1Graph TheoryIradmusa
Chromatic number of $\frac{3}{3}$-power of graph✭✭0Graph TheoryIradmusa
Polignac's Conjecturede Polignac✭✭✭0Number TheoryHugh Barker
Three-chromatic (0,2)-graphsPayan✭✭0Graph Theory » ColoringGordon Royle
Equality in a matroidal circumference boundOxley; Royle✭✭0Combinatorics » Matroid TheoryGordon Royle
Odd cycles and low oddness✭✭0Graph TheoryGagik
Oriented trees in n-chromatic digraphsBurr✭✭✭0Graph Theory » Directed Graphsfhavet
Decomposing an even tournament in directed paths.Alspach; Mason; Pullman✭✭✭0Graph Theory » Directed Graphs » Tournamentsfhavet
Antidirected trees in digraphsAddario-Berry; Havet; Linhares Sales; Reed; Thomassé✭✭0Graph Theory » Directed Graphsfhavet
Directed path of length twice the minimum outdegreeThomassé✭✭✭0Graph Theory » Directed Graphsfhavet
Caccetta-Häggkvist ConjectureCaccetta; Häggkvist✭✭✭✭0Graph Theory » Directed Graphsfhavet
Ádám's ConjectureÁdám✭✭✭0Graph Theory » Directed Graphsfhavet
Stable set meeting all longest directed paths.Laborde; Payan; Xuong N.H.✭✭0Graph Theoryfhavet
Splitting a digraph with minimum outdegree constraintsAlon✭✭✭0Graph Theory » Directed Graphsfhavet
Long directed cycles in diregular digraphsJackson✭✭✭0Graph Theory » Directed Graphsfhavet
Strong edge colouring conjectureErdos; Nesetril✭✭0Graph Theory » Coloring » Edge coloringfhavet

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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