Noel, Jonathan A.

Graph Theory » Extremal G.T.

Weak saturation of the cube in the clique ★

Author(s): Morrison; Noel

Problem

Determine $\text{wsat}(K_n,Q_3)$ .

Keywords: bootstrap percolation; hypercube; Weak saturation

Posted by Jon Noel
updated April 6th, 2016

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Combinatorics

Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★

Author(s): Morrison; Noel

Problem Determine the smallest percolating set for the $4$ -neighbour bootstrap process in the hypercube.

Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation

Posted by Jon Noel
updated September 20th, 2015

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Combinatorics

Saturation in the Hypercube ★★

Author(s): Morrison; Noel; Scott

Question What is the saturation number of cycles of length $2\ell$ in the $d$ -dimensional hypercube?

Keywords: cycles; hypercube; minimum saturation; saturation

Posted by Jon Noel
updated September 20th, 2015

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Graph Theory » Coloring » Vertex coloring

Cycles in Graphs of Large Chromatic Number ★★

Author(s): Brewster; McGuinness; Moore; Noel

Conjecture If $\chi(G)>k$ , then $G$ contains at least $\frac{(k+1)(k-1)!}{2}$ cycles of length $0\bmod k$ .

Keywords: chromatic number; cycles

Posted by Jon Noel
updated September 20th, 2015

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Combinatorics » Posets

Saturated $k$-Sperner Systems of Minimum Size ★★

Author(s): Morrison; Noel; Scott

Question Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\geq n_0(k)$ , then every saturated $k$ -Sperner system $\mathcal{F}\subseteq \mathcal{P}(X)$ has cardinality at least $2^{(1+o(1))ck}$ ?

Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system

Posted by Jon Noel
updated April 12th, 2014

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Geometry

Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by projective plane we mean the set of all lines through the origin in $\mathbb{R}^3$ .

Definition Say that a subset $S$ of the projective plane is octahedral if all lines in $S$ pass through the closure of two opposite faces of a regular octahedron centered at the origin.

Definition Say that a subset $S$ of the projective plane is weakly octahedral if every set $S'\subseteq S$ such that $|S'|=3$ is octahedral.

Conjecture Suppose that the projective plane can be partitioned into four sets, say $S_1,S_2,S_3$ and $S_4$ such that each set $S_i$ is weakly octahedral. Then each $S_i$ is octahedral.

Keywords: Partitioning; projective plane

Posted by Jon Noel
updated August 27th, 2013

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Graph Theory » Coloring » Vertex coloring

Choosability of Graph Powers ★★

Author(s): Noel

Question (Noel, 2013) Does there exist a function $f(k)=o(k^2)$ such that for every graph $G$ , $\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?$