![](/files/happy5.png)
Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation
Saturation in the Hypercube ★★
Author(s): Morrison; Noel; Scott
![$ 2\ell $](/files/tex/e6160c4357fdf2ec5854a3cc78837f8a67caa5c5.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
Keywords: cycles; hypercube; minimum saturation; saturation
Cycles in Graphs of Large Chromatic Number ★★
Author(s): Brewster; McGuinness; Moore; Noel
![$ \chi(G)>k $](/files/tex/84d787e716a616f1d9b6d33aea0d9f0777cb1df3.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \frac{(k+1)(k-1)!}{2} $](/files/tex/2576c24e6815c0bf97ab23f18ad24cf5421aeac4.png)
![$ 0\bmod k $](/files/tex/ef4d29155ecd56ddbfea81561d000d0e5823edb7.png)
Keywords: chromatic number; cycles
The Double Cap Conjecture ★★
Author(s): Kalai
![$ \mathbb{R}^n $](/files/tex/2010c953180b3521ec2f66d10e1f40ec71d44574.png)
![$ \pi/4 $](/files/tex/01608ea3b80f85b77096d16610a43e184782386c.png)
Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere
Circular flow numbers of $r$-graphs ★★
Author(s): Steffen
A nowhere-zero -flow
on
is an orientation
of
together with a function
from the edge set of
into the real numbers such that
, for all
, and
.
A -regular graph
is a
-graph if
for every
with
odd.
![$ t > 1 $](/files/tex/13690a4a2acccbf67e32711965c98ee5b60756eb.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ (2t+1) $](/files/tex/da4d60d92c98256f762bb69398437f3914ef0fa6.png)
![$ F_c(G) \leq 2 + \frac{2}{t} $](/files/tex/35f6f6ba01e2a1f8f2c867888c086731d735cc74.png)
Keywords: flow conjectures; nowhere-zero flows
Circular flow number of regular class 1 graphs ★★
Author(s): Steffen
A nowhere-zero -flow
on
is an orientation
of
together with a function
from the edge set of
into the real numbers such that
, for all
, and
. The circular flow number of
is inf
has a nowhere-zero
-flow
, and it is denoted by
.
A graph with maximum vertex degree is a class 1 graph if its edge chromatic number is
.
![$ t \geq 1 $](/files/tex/f9082ade09146d7aa9994735ba4ad788d0583b0c.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ (2t+1) $](/files/tex/da4d60d92c98256f762bb69398437f3914ef0fa6.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ F_c(G) \leq 2 + \frac{2}{t} $](/files/tex/35f6f6ba01e2a1f8f2c867888c086731d735cc74.png)