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Let $G$ be a graph and $m,n\in \mathbb{N}$ . The graph $G^{\frac{m}{n}}$ is defined to be the $m$ -power of the $n$ -subdivision of $G$ . In other words, $G^{\frac{m}{n}}=(G^{\frac{1}{n}})^m$ .

Conjecture Let $G$ be a graph with $\Delta(G)\geq 2$ . Then $\chi(G^{\frac{3}{3}})\leq 2\Delta(G)+1$ .

Keywords:

Posted by Iradmusa
updated April 20th, 2023

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Graph Theory

3-Edge-Coloring Conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture Suppose $G$ with $|V(G)|>2$ is a connected cubic graph admitting a $3$ -edge coloring. Then there is an edge $e \in E(G)$ such that the cubic graph homeomorphic to $G-e$ has a $3$ -edge coloring.

Keywords: 3-edge coloring; 4-flow; removable edge

Posted by arthur
updated June 23rd, 2022

4 comments

Graph Theory » Basic G.T. » Cycles

r-regular graphs are not uniquely hamiltonian. ★★★

Author(s): Sheehan

Conjecture If $G$ is a finite $r$ -regular graph, where $r > 2$ , then $G$ is not uniquely hamiltonian.

Keywords: hamiltonian; regular; uniquely hamiltonian

Posted by Robert Samal
updated June 20th, 2022

3 comments

Geometry

Partition of Complete Geometric Graph into Plane Trees ★★

Author(s):

Conjecture Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.

Keywords: complete geometric graph, edge colouring

Posted by David Wood
updated January 6th, 2022

1 comment

Topology

Smooth 4-dimensional Poincare conjecture ★★★★

Author(s): Poincare; Smale; Stallings

Conjecture If a $4$ -manifold has the homotopy type of the $4$ -sphere $S^4$ , is it diffeomorphic to $S^4$ ?

Keywords: 4-manifold; poincare; sphere

Posted by rybu
updated August 26th, 2021

1 comment

Graph Theory

Book Thickness of Subdivisions ★★

Author(s): Blankenship; Oporowski

Let $G$ be a finite undirected simple graph.

A $k$ -page book embedding of $G$ consists of a linear order $\preceq$ of $V(G)$ and a (non-proper) $k$ -colouring of $E(G)$ such that edges with the same colour do not cross with respect to $\preceq$ . That is, if $v\prec x\prec w\prec y$ for some edges $vw,xy\in E(G)$ , then $vw$ and $xy$ receive distinct colours.

One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.

The book thickness of $G$ , denoted by bt $(G)$ is the minimum integer $k$ for which there is a $k$ -page book embedding of $G$ .

Let $G'$ be the graph obtained by subdividing each edge of $G$ exactly once.

Conjecture There is a function $f$ such that for every graph $G$ , $\text{bt}(G) \leq f( \text{bt}(G') )\enspace.$

Keywords: book embedding; book thickness

Posted by David Wood
updated July 10th, 2021

1 comment

Number Theory

Primitive pythagorean n-tuple tree ★★

Author(s):

Conjecture Find linear transformation construction of primitive pythagorean n-tuple tree!

Keywords:

Posted by tsihonglau
updated June 16th, 2021

1 comment

Geometry » Algebraic Geometry

Jacobian Conjecture ★★★

Author(s): Keller

Conjecture Let $k$ be a field of characteristic zero. A collection $f_1,\ldots,f_n$ of polynomials in variables $x_1,\ldots,x_n$ defines an automorphism of $k^n$ if and only if the Jacobian matrix is a nonzero constant.

Keywords: Affine Geometry; Automorphisms; Polynomials

Posted by Charles
updated March 4th, 2021

1 comment

Topology

Inscribed Square Problem ★★

Author(s): Toeplitz

Conjecture Does every Jordan curve have 4 points on it which form the vertices of a square?

Keywords: simple closed curve; square

Posted by dlh12
updated February 25th, 2021

8 comments

Graph Theory » Basic G.T.

Complete bipartite subgraphs of perfect graphs ★★

Author(s): Fox

Problem Let $G$ be a perfect graph on $n$ vertices. Is it true that either $G$ or $\bar{G}$ contains a complete bipartite subgraph with bipartition $(A,B)$ so that $|A|, |B| \ge n^{1 - o(1)}$ ?

Keywords: perfect graph

Posted by mdevos
updated January 19th, 2021

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Chords of longest cycles ★★★

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Chromatic number of $\frac{3}{3}$-power of graph ★★

3-Edge-Coloring Conjecture ★★★

r-regular graphs are not uniquely hamiltonian. ★★★

Partition of Complete Geometric Graph into Plane Trees ★★

Smooth 4-dimensional Poincare conjecture ★★★★

Book Thickness of Subdivisions ★★

Primitive pythagorean n-tuple tree ★★

Jacobian Conjecture ★★★

Inscribed Square Problem ★★

Complete bipartite subgraphs of perfect graphs ★★

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