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Thomassen, Carsten
Partitionning a tournament into k-strongly connected subtournaments. ★★
Author(s): Thomassen
Problem Let
be positve integer Does there exists an integer
such that every
-strong tournament
admits a partition
of its vertex set such that the subtournament induced by
is a non-trivial
-strong for all
.
![$ k_1, \dots , k_p $](/files/tex/dd32073e76a3e937a33f354d483a622b518fd952.png)
![$ g(k_1, \dots , k_p) $](/files/tex/d421e344ef4e58f862c849a1510c5bfbc987695c.png)
![$ g(k_1, \dots , k_p) $](/files/tex/d421e344ef4e58f862c849a1510c5bfbc987695c.png)
![$ T $](/files/tex/79f55d2e1d83a7726c807a70cbe756713b0437b6.png)
![$ (V_1\dots , V_p) $](/files/tex/5a9d8fc043fbbf884f7a132e075c21bcfc070b50.png)
![$ V_i $](/files/tex/af854be1f03aac481e0a165c3908976d4b5b0aa0.png)
![$ k_i $](/files/tex/e4854627e64b06bb06bbeb46f57f3b1e9b30b1b7.png)
![$ 1\leq i\leq p $](/files/tex/4e9f329cd88669519e011cd4cd2fb9a90b5b4828.png)
Keywords:
Edge-disjoint Hamilton cycles in highly strongly connected tournaments. ★★
Author(s): Thomassen
Conjecture For every
, there is an integer
so that every strongly
-connected tournament has
edge-disjoint Hamilton cycles.
![$ k\geq 2 $](/files/tex/13bc863dca3c6b96ebfd2de373f0fe820c58b62b.png)
![$ f(k) $](/files/tex/e055b3867e7cb3cc4b2f50739eedda7657999214.png)
![$ f(k) $](/files/tex/e055b3867e7cb3cc4b2f50739eedda7657999214.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
Keywords:
Subgraph of large average degree and large girth. ★★
Author(s): Thomassen
Conjecture For all positive integers
and
, there exists an integer
such that every graph of average degree at least
contains a subgraph of average degree at least
and girth greater than
.
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
Keywords:
Arc-disjoint out-branching and in-branching ★★
Author(s): Thomassen
Conjecture There exists an integer
such that every
-arc-strong digraph
with specified vertices
and
contains an out-branching rooted at
and an in-branching rooted at
which are arc-disjoint.
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ D $](/files/tex/b8653a25aff72e3dacd3642492c24c2241f0058c.png)
![$ u $](/files/tex/06183efdad837019eb0937c4e40f9e7beaa2e8d8.png)
![$ v $](/files/tex/96cbd9a16c6a5eab03815b093b08f3b2db614e9a.png)
![$ u $](/files/tex/06183efdad837019eb0937c4e40f9e7beaa2e8d8.png)
![$ v $](/files/tex/96cbd9a16c6a5eab03815b093b08f3b2db614e9a.png)
Keywords:
Counting 3-colorings of the hex lattice ★★
Author(s): Thomassen
Problem Find
.
![$ \lim_{n \rightarrow \infty} (\chi( H_n , 3)) ^{ 1 / |V(H_n)| } $](/files/tex/f0a8cb3e30752d801fe1d52a9759cf0e47894c8a.png)
Keywords: coloring; Lieb's Ice Constant; tiling; torus
Chords of longest cycles ★★★
Author(s): Thomassen
Conjecture If
is a 3-connected graph, every longest cycle in
has a chord.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: chord; connectivity; cycle
The Bermond-Thomassen Conjecture ★★
Conjecture For every positive integer
, every digraph with minimum out-degree at least
contains
disjoint cycles.
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ 2k-1 $](/files/tex/7dee0e3b6c6462122fea33090b927b7d6e36817a.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
Keywords: cycles
Hamiltonian cycles in line graphs ★★★
Author(s): Thomassen
Conjecture Every 4-connected line graph is hamiltonian.
Keywords: hamiltonian; line graphs
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