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Chords of longest cycles ★★★
Author(s): Thomassen
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: chord; connectivity; cycle
Do any three longest paths in a connected graph have a vertex in common? ★★
Author(s): Gallai
Keywords:
Chromatic number of $\frac{3}{3}$-power of graph ★★
Author(s):
Let be a graph and
. The graph
is defined to be the
-power of the
-subdivision of
. In other words,
.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \Delta(G)\geq 2 $](/files/tex/25c622d02e81b413d70fba0eb49b1ca09ad2ee1b.png)
![$ \chi(G^{\frac{3}{3}})\leq 2\Delta(G)+1 $](/files/tex/77f35e23639197d1972bd2a477a135f33ef44731.png)
Keywords:
3-Edge-Coloring Conjecture ★★★
Author(s): Arthur; Hoffmann-Ostenhof
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ |V(G)|>2 $](/files/tex/ac99a71a6e28acc3052e542089238e810d347be6.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ e \in E(G) $](/files/tex/730c5d64c8d749c640adc18eb493c641ff1addc9.png)
![$ G-e $](/files/tex/a9c40841d6043b14ca9501d156e86164ad3f81e5.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
Keywords: 3-edge coloring; 4-flow; removable edge
r-regular graphs are not uniquely hamiltonian. ★★★
Author(s): Sheehan
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
![$ r > 2 $](/files/tex/97a6c1471f5d46c5d21bf3e382139df59f21bd80.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: hamiltonian; regular; uniquely hamiltonian
Partition of Complete Geometric Graph into Plane Trees ★★
Author(s):
Keywords: complete geometric graph, edge colouring
Smooth 4-dimensional Poincare conjecture ★★★★
Author(s): Poincare; Smale; Stallings
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
![$ S^4 $](/files/tex/8973308b8ba6ed78524b0e4751ab814bbaaa57e2.png)
![$ S^4 $](/files/tex/8973308b8ba6ed78524b0e4751ab814bbaaa57e2.png)
Keywords: 4-manifold; poincare; sphere
Book Thickness of Subdivisions ★★
Author(s): Blankenship; Oporowski
Let be a finite undirected simple graph.
A -page book embedding of
consists of a linear order
of
and a (non-proper)
-colouring of
such that edges with the same colour do not cross with respect to
. That is, if
for some edges
, then
and
receive distinct colours.
One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.
The book thickness of , denoted by bt
is the minimum integer
for which there is a
-page book embedding of
.
Let be the graph obtained by subdividing each edge of
exactly once.
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$$ \text{bt}(G) \leq f( \text{bt}(G') )\enspace. $$](/files/tex/0aedb9a9a46520e4c6b7a352924b0c2e2fc329d6.png)
Keywords: book embedding; book thickness
Primitive pythagorean n-tuple tree ★★
Author(s):
Keywords:
Jacobian Conjecture ★★★
Author(s): Keller
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ f_1,\ldots,f_n $](/files/tex/c4cbeed48b0c6ce2a6294c75261da86958fb7393.png)
![$ x_1,\ldots,x_n $](/files/tex/aaa29ecce30762f5fa8655c6d5de69088530a526.png)
![$ k^n $](/files/tex/b2baee4a1dacdb8cbb7880c33194a814fe14947b.png)
Keywords: Affine Geometry; Automorphisms; Polynomials
Inscribed Square Problem ★★
Author(s): Toeplitz
Keywords: simple closed curve; square
Complete bipartite subgraphs of perfect graphs ★★
Author(s): Fox
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \bar{G} $](/files/tex/1e3dac00cfbbb43d9181a85bc8a5389e16490552.png)
![$ (A,B) $](/files/tex/e0150654d2f6aa6b2ea50e726af4747878a605fc.png)
![$ |A|, |B| \ge n^{1 - o(1)} $](/files/tex/457f19bbb97576457ac692bc6acdd4ac8e9f39a1.png)
Keywords: perfect graph
Transversal achievement game on a square grid ★★
Author(s): Erickson
![$ n \times n $](/files/tex/fb4b8a31555841e933e45cae91f49adc3b187f2f.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
Keywords: game
Graceful Tree Conjecture ★★★
Author(s):
Keywords: combinatorics; graceful labeling
Extremal problem on the number of tree endomorphism ★★
Author(s): Zhicong Lin
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
Keywords:
3-Colourability of Arrangements of Great Circles ★★
Author(s): Felsner; Hurtado; Noy; Streinu
Consider a set of great circles on a sphere with no three circles meeting at a point. The arrangement graph of
has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
Keywords: arrangement graph; graph coloring
KPZ Universality Conjecture ★★★
Author(s):
Keywords: KPZ equation, central limit theorem
Friendly partitions ★★
Author(s): DeVos
A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
Finite entailment of Positive Horn logic ★★
Author(s): Martin
![$ \exists, \forall, \wedge, = $](/files/tex/daf2de7da53fbeafe84535aa54eb026a1b767f3d.png)
![$ pH \wedge \neg pH $](/files/tex/1483dab6c9b23872227df3bb1f2a487c3f552f34.png)
Keywords: entailment; finite satisfiability; horn logic
Triangle free strongly regular graphs ★★★
Author(s):
Keywords: strongly regular; triangle free