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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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Geometry

Convex Equipartitions with Extreme Perimeter ★★

Author(s): Nandakumar

To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.

Remark: It appears maximizing the total perimeter is the easier problem.

Keywords: convex equipartition

Posted by Nandakumar
updated October 22nd, 2015

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Combinatorics

Turán Problem for $10$-Cycles in the Hypercube ★★

Author(s): Erdos

Problem Bound the extremal number of $C_{10}$ in the hypercube.

Keywords: cycles; extremal combinatorics; hypercube

Posted by Jon Noel
updated September 20th, 2015

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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

The Double Cap Conjecture ★★

Author(s): Kalai

Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of $\mathbb{R}^n$ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $\pi/4$ around the north and south poles.

Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere

Posted by Jon Noel
updated September 15th, 2015

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Graph Theory

Circular flow numbers of $r$-graphs ★★

Author(s): Steffen

A nowhere-zero $r$ -flow $(D(G),\phi)$ on $G$ is an orientation $D$ of $G$ together with a function $\phi$ from the edge set of $G$ into the real numbers such that $1 \leq |\phi(e)| \leq r-1$ , for all $e \in E(G)$ , and $\sum_{e \in E^+(v)}\phi(e) = \sum_{e \in E^-(v)}\phi(e), \textrm{ for all } v \in V(G)$ .

A $(2t+1)$ -regular graph $G$ is a $(2t+1)$ -graph if $|\partial_G(X)| \geq 2t+1$ for every $X \subseteq V(G)$ with $|X|$ odd.

Conjecture Let $t > 1$ be an integer. If $G$ is a $(2t+1)$ -graph, then $F_c(G) \leq 2 + \frac{2}{t}$ .

Keywords: flow conjectures; nowhere-zero flows

Posted by Eckhard Steffen
updated August 6th, 2015

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Graph Theory » Coloring » Nowhere-zero flows

Circular flow number of regular class 1 graphs ★★

Author(s): Steffen

A graph with maximum vertex degree $k$ is a class 1 graph if its edge chromatic number is $k$ .

Conjecture Let $t \geq 1$ be an integer and $G$ a $(2t+1)$ -regular graph. If $G$ is a class 1 graph, then $F_c(G) \leq 2 + \frac{2}{t}$ .

Keywords: nowhere-zero flow, edge-colorings, regular graphs

Posted by Eckhard Steffen
updated August 5th, 2015

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Geometry

Chromatic number of associahedron ★★

Author(s): Fabila-Monroy; Flores-Penaloza; Huemer; Hurtado; Urrutia; Wood

Conjecture Associahedra have unbounded chromatic number.

Keywords: associahedron, graph colouring, chromatic number

Posted by David Wood
updated June 2nd, 2015

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Number Theory » Analytic N.T.

Are there infinite number of Mersenne Primes? ★★★★

Author(s):

Conjecture A Mersenne prime is a Mersenne number $M_n = 2^p - 1$ that is prime.

Are there infinite number of Mersenne Primes?

Keywords: Mersenne number; Mersenne prime

Posted by kurtulmehtap
updated February 4th, 2015

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Number Theory » Analytic N.T.

Are all Mersenne Numbers with prime exponent square-free? ★★★

Author(s):

Conjecture Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free?

Keywords: Mersenne number

Posted by kurtulmehtap
updated February 4th, 2015

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Topology

What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

Let $\mathfrak{A}$ be an indexed family of sets.

Products are $\prod A$ for $A \in \prod \mathfrak{A}$ .

Hyperfuncoids are filters $\mathfrak{F} \Gamma$ on the lattice $\Gamma$ of all finite unions of products.

Problem Is $\bigcap^{\mathsf{\tmop{FCD}}}$ a bijection from hyperfuncoids $\mathfrak{F} \Gamma$ to:

$\mathfrak{A}$

If yes, is $\operatorname{up}^{\Gamma}$ defining the inverse bijection? If not, characterize the image of the function $\bigcap^{\mathsf{\tmop{FCD}}}$ defined on $\mathfrak{F} \Gamma$ .

Consider also the variant of this problem with the set $\Gamma$ replaced with the set $\Gamma^{\ast}$ of complements of elements of the set $\Gamma$ .

Keywords: hyperfuncoids; multidimensional

Posted by porton
updated December 9th, 2014

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Topology

Another conjecture about reloids and funcoids ★★

Author(s): Porton

Definition $\square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f$ for reloid $f$ .

Conjecture $(\mathsf{RLD})_{\Gamma} f = \square (\mathsf{RLD})_{\mathrm{in}} f$ for every funcoid $f$ .

Note: it is known that $(\mathsf{RLD})_{\Gamma} f \ne \square (\mathsf{RLD})_{\mathrm{out}} f$ (see below mentioned online article).

Keywords:

Posted by porton
updated November 28th, 2014

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Analysis

Inequality for square summable complex series ★★

Author(s): Retkes

Conjecture For all $\alpha=(\alpha_1,\alpha_2,\ldots)\in l_2(\cal{C})$ the following inequality holds $\sum_{n\geq 1}|\alpha_n|^2\geq \frac{6}{\pi^2}\sum_{k\geq0}\bigg| \sum_{l\geq0}\frac{1}{l+1}\alpha_{2^k(2l+1)}\bigg|^2$

Keywords: Inequality

Posted by tigris35711
updated October 29th, 2014

2 comments

Theoretical Comp. Sci. » Complexity

One-way functions exist ★★★★

Author(s):

Conjecture One-way functions exist.

Keywords: one way function

Posted by porton
updated October 12th, 2014

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Graph Theory

Chromatic Number of Common Graphs ★★

Author(s): Hatami; Hladký; Kráľ; Norine; Razborov

Question Do common graphs have bounded chromatic number?

Keywords: common graph

Posted by David Wood
updated August 15th, 2014

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Number Theory

Erdős–Straus conjecture ★★

Author(s): Erdos; Straus

Conjecture

For all $n > 2$ , there exist positive integers $x$ , $y$ , $z$ such that $1/x + 1/y + 1/z = 4/n$ .

Keywords: Egyptian fraction

Posted by ACW
updated July 14th, 2014

4 comments

Number Theory » Combinatorial N.T.

The 3n+1 conjecture ★★★

Author(s): Collatz

Conjecture Let $f(n) = 3n+1$ if $n$ is odd and $\frac{n}{2}$ if $n$ is even. Let $f(1) = 1$ . Assume we start with some number $n$ and repeatedly take the $f$ of the current number. Prove that no matter what the initial number is we eventually reach $1$ .