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List Total Colouring Conjecture ★★
Author(s): Borodin; Kostochka; Woodall
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \chi_\ell(G)=\chi(G) $](/files/tex/0a2573f7d1a57016f919f018635cd3f9f9875fc4.png)
Keywords: list coloring; Total coloring; total graphs
Partitioning the Projective Plane ★★
Author(s): Noel
Throughout this post, by projective plane we mean the set of all lines through the origin in .
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S'\subseteq S $](/files/tex/e8109a0f5c1dcb11fe3245b53b8bd2bc9d6418d1.png)
![$ |S'|=3 $](/files/tex/e1b8bc4df405ab37c6b4aa2f638a5c2df882f7a9.png)
![$ S_1,S_2,S_3 $](/files/tex/6f2284a502ee75f719fa3d5c2430c467e11df0c4.png)
![$ S_4 $](/files/tex/a0fc8ce0b0dfbf88309c7c045fff90a5cadd5117.png)
![$ S_i $](/files/tex/110ae457d97eebe47aa4d2e8c6237fdb9317f11e.png)
![$ S_i $](/files/tex/110ae457d97eebe47aa4d2e8c6237fdb9317f11e.png)
Keywords: Partitioning; projective plane
Kriesell's Conjecture ★★
Author(s): Kriesell
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ T\subseteq V(G) $](/files/tex/0acf1a8ecf3a0737d34c34b8652d10a2c33df19b.png)
![$ u,v\in T $](/files/tex/bbcef09f86563651f02daa6bbae826055f48edfb.png)
![$ 2k $](/files/tex/bded1a5bf39ed2baaf98bd8c04cea4667dd89b58.png)
![$ u $](/files/tex/06183efdad837019eb0937c4e40f9e7beaa2e8d8.png)
![$ v $](/files/tex/96cbd9a16c6a5eab03815b093b08f3b2db614e9a.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ T $](/files/tex/79f55d2e1d83a7726c807a70cbe756713b0437b6.png)
Keywords: Disjoint paths; edge-connectivity; spanning trees
2-colouring a graph without a monochromatic maximum clique ★★
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ 5 $](/files/tex/87f5fe1d4b06035debb52cf2d67802fbfa9cb4ab.png)
![$ 2 $](/files/tex/5271e36bb1c040e0f14061d89cd97d0c86d4e06f.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: maximum clique; Partitioning
Almost all non-Hamiltonian 3-regular graphs are 1-connected ★★
Author(s): Haythorpe
![$ NH(n) $](/files/tex/dc7c2bcdf0fdc40cfaa4e4b35d12bfa84042b5f4.png)
![$ 2n $](/files/tex/56259815f2fdf87e92dd22e0058206e8e20fb986.png)
![$ NHB(n) $](/files/tex/f397fa463c02be6928f5ad6655b68aa9a62c7195.png)
![$ 2n $](/files/tex/56259815f2fdf87e92dd22e0058206e8e20fb986.png)
Is it true that ?
Erdős–Faber–Lovász conjecture ★★★
Author(s): Erdos; Faber; Lovasz
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
Keywords: chromatic number
Are there only finite Fermat Primes? ★★★
Author(s):
![\[ F_n = 2^{2^n } + 1 \]](/files/tex/0da5a50010e4e5df91c0d58080245ece34ec9ca6.png)
Keywords:
Are all Fermat Numbers square-free? ★★★
Author(s):
![\[ F_n = 2^{2^{n } } + 1 \]](/files/tex/70ca73d7e82af2fee084a8417e172c58cf78b376.png)
Keywords:
Choosability of Graph Powers ★★
Author(s): Noel
![$ f(k)=o(k^2) $](/files/tex/bd642e5dd66f1577cedf5fed57f75187a80168ac.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![\[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\]](/files/tex/989db06683633e86605c26e7d9f0bffc7e46a496.png)
Keywords: choosability; chromatic number; list coloring; square of a graph
Erdős-Posa property for long directed cycles ★★
![$ \ell \geq 2 $](/files/tex/061b4d40c28de2de3ccfebdeeb52a6730a9cee76.png)
![$ n\geq 0 $](/files/tex/6a9c6e677c8a074e2f8306290b3acb959626d2a4.png)
![$ t_n=t_n(\ell) $](/files/tex/2ce8e7dd4d72ec3247e3b656a3e3cfa985241df5.png)
![$ D $](/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png)
![$ D $](/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
![$ T $](/files/tex/79f55d2e1d83a7726c807a70cbe756713b0437b6.png)
![$ t_n $](/files/tex/4e1d881c711abbf6021a83fe432d28abc22d717b.png)
![$ D-T $](/files/tex/15b90efb84c80a9728144fa55150c691b4230a8a.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
Keywords:
Large acyclic induced subdigraph in a planar oriented graph. ★★
Author(s): Harutyunyan
![$ D $](/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png)
![$ \frac{3}{5} |V(D)| $](/files/tex/5a32f551bf4043e68e6d6a234d5683ce8fff1aea.png)
Keywords:
Polignac's Conjecture ★★★
Author(s): de Polignac
In particular, this implies:
Alexa's Conjecture on Primality ★★
Author(s): Alexa
![$ r_i $](/files/tex/bd013755b6effdbb49a2ee8dcec9023b56000a4a.png)
![$ p\in\mathbb{N} $](/files/tex/5214316bdda9be56a163d54dbebb20eeb3f1a0d6.png)
![$ p \ge 8 $](/files/tex/2564938a26a5c2d5710bc0f5817232309dc67929.png)
![$$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$](/files/tex/99af565f4cc4d3bab11eb3fbf54f78626678d484.png)
Keywords: primality
P vs. BPP ★★★
Author(s): Folklore
Keywords: BPP; circuit complexity; pseudorandom generators
Goldbach conjecture ★★★★
Author(s): Goldbach
Keywords: additive basis; prime
Goldberg's conjecture ★★★
Author(s): Goldberg
The overfull parameter is defined as follows:
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \chi'(G) \le \max\{ \Delta(G) + 1, w(G) \} $](/files/tex/688f47d207abb99bb15f8fc2353553cf3904fc1d.png)
Keywords: edge-coloring; multigraph
Cyclic spanning subdigraph with small cyclomatic number ★★
Author(s): Bondy
![$ D $](/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png)
![$ D $](/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png)
![$ \alpha(D) $](/files/tex/d1e0ce4240c9d4761bc744654b19250832ac5a91.png)
Keywords:
inverse of an integer matrix ★★
Author(s): Gregory
![$ \ge 2 $](/files/tex/03e64a95c7ce55689a656134f423984ca6f051c3.png)
![$ (m \le n) $](/files/tex/3b3dea6d64446f720ff1266bd875040e8eb8065f.png)
Keywords: invertable matrices, integer matrices
Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament ★★
Author(s): Yuster
![$ T $](/files/tex/79f55d2e1d83a7726c807a70cbe756713b0437b6.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ \left \lceil n(n-1)/6 - n/3\right\rceil $](/files/tex/559f44f15c03e6e3ad0ebeafb9428b5563c17d28.png)
Keywords:
Arc-disjoint directed cycles in regular directed graphs ★★
Author(s): Alon; McDiarmid; Molloy
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ {k+1 \choose 2} $](/files/tex/783a85f7ae121a116b289d0f5b090f98fc9f959e.png)
Keywords: