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Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems) ★★★★
Author(s):
![$ Diff^{r}(M) $](/files/tex/96b7458da5816649d7f2ad399ef03fb416359d46.png)
![$ C^{r} $](/files/tex/b66404b2329b94aa018acb2481e504cd18fcd638.png)
![$ \chi^{r}(M) $](/files/tex/60a0032cb74d2f8ef538590ff6e92abecc056eec.png)
![$ C^{r} $](/files/tex/b66404b2329b94aa018acb2481e504cd18fcd638.png)
![$ D\subset Diff^{r}(M) $](/files/tex/ebb61bcfdb969ad1b3cfd893afb858ee13d86f5a.png)
![$ D\subset \chi^{r}(M) $](/files/tex/a9b664fbb7fb5dd089d5f67b082350ac8f9cef3f.png)
![$ \forall f\in D $](/files/tex/4a046aee22105c97503f4ec3488a0dcb4dcdc75f.png)
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .
Keywords: Attractors , basins, Finite
Closing Lemma for Diffeomorphism (Dynamical Systems) ★★★★
Author(s): Charles Pugh
![$ f\in Diff^{r}(M) $](/files/tex/d45758dc65c33e4e1d18e889906d62017e3ad4b8.png)
![$ p\in\omega_{f} $](/files/tex/6895bfe444c3ed2420744f8fa4a2244580943023.png)
![$ V_{f}\subset Diff^{r}(M) $](/files/tex/5068fd3c71b626b7da0dc70cd04a36131d00be24.png)
![$ g\in V_{f} $](/files/tex/b17a7c9db99af8d718bac041b728beda5e26c111.png)
![$ p $](/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for
. In the case of Flows this has been solved by Sushei Hayahshy for
. But in the two cases the problem is wide open for
Keywords: Dynamics , Pertubation
Sub-atomic product of funcoids is a categorical product ★★
Author(s):
- \item Product morphism is defined similarly to the category of topological spaces. \item Product object is the sub-atomic product. \item Projections are sub-atomic projections.
See details, exact definitions, and attempted proofs here.
Keywords:
Bounding the on-line choice number in terms of the choice number ★★
Author(s): Zhu
![$ \text{ch}^{\text{OL}}-\text{ch} $](/files/tex/3aa1320c57f076f184f0f05f44e350d3ffea4fa4.png)
Keywords: choosability; list coloring; on-line choosability
Are almost all graphs determined by their spectrum? ★★★
Author(s):
Keywords: cospectral; graph invariant; spectrum
Signing a graph to have small magnitude eigenvalues ★★
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ A $](/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png)
![$ +1 $](/files/tex/155fca3c17d66548c323f203be786f9387842fe4.png)
![$ -1 $](/files/tex/26833acbe5abb13c40595cebdee81f595c59a397.png)
![$ 2 \sqrt{d-1} $](/files/tex/40581c05b66d632e7bdb2bcb852e63443663853a.png)
Keywords: eigenvalue; expander; Ramanujan graph; signed graph; signing
The Bollobás-Eldridge-Catlin Conjecture on graph packing ★★★
Author(s):
![$ G_1 $](/files/tex/1475475906abb943f311289729184c527071d32f.png)
![$ G_2 $](/files/tex/7c390ecd91deb4948e85912eba8f5594f03ea2f4.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ (\Delta(G_1) + 1) (\Delta(G_2) + 1) < n + 1 $](/files/tex/949ceded82df0b3fa9d75e8d2ae19a95c5ed832d.png)
![$ G_1 $](/files/tex/1475475906abb943f311289729184c527071d32f.png)
![$ G_2 $](/files/tex/7c390ecd91deb4948e85912eba8f5594f03ea2f4.png)
Keywords: graph packing
Decomposing k-arc-strong tournament into k spanning strong digraphs ★★
Author(s): Bang-Jensen; Yeo
Keywords:
PTAS for feedback arc set in tournaments ★★
Keywords: feedback arc set; PTAS; tournament
Partitionning a tournament into k-strongly connected subtournaments. ★★
Author(s): Thomassen
![$ 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 $](/files/tex/4e9f329cd88669519e011cd4cd2fb9a90b5b4828.png)
Keywords:
Weighted colouring of hexagonal graphs. ★★
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ p:V(G)\rightarrow \mathbb{N} $](/files/tex/6b14cbdc1cda96520fd87d9507b262f943ccf8ba.png)
![$$\chi(G,p) \leq \frac{9}{8}\omega(G,p) + c $$](/files/tex/9b4ccf7620c2a90a97c40177afd9201723864ba8.png)
Keywords:
Colouring the square of a planar graph ★★
Author(s): Wegner
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \Delta $](/files/tex/e3f8e135c571143e94f1d4f236326b862080b200.png)
- \item at most
![$ 7 $](/files/tex/daebc9d78b81a4cd1f0e6bae70785e92cf2ce1ea.png)
![$ \Delta =3 $](/files/tex/aced9ebeab44df5a1f3efd1bfb967dd0fe0883c4.png)
![$ \Delta+5 $](/files/tex/a7be33e67abdf15a956bdc393747bb7f75bbd905.png)
![$ 4\leq\Delta\leq 7 $](/files/tex/edab53b759e15d01350d22f54590e6ee9a99bca9.png)
![$ \left\lfloor\frac32\,\Delta\right\rfloor+1 $](/files/tex/d5aa6f60013e17601f35a898a36936e347bc9a8b.png)
![$ \Delta\ge8 $](/files/tex/0782472fc66f76ac853e14ef0f2ded9aaac96451.png)
Keywords:
List chromatic number and maximum degree of bipartite graphs ★★
Author(s): Alon
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \Delta $](/files/tex/e3f8e135c571143e94f1d4f236326b862080b200.png)
![$ c \log \Delta $](/files/tex/84aaddf436e9bd784cec262225dc1b22798556fd.png)
Keywords:
Hamilton decomposition of prisms over 3-connected cubic planar graphs ★★
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
Keywords:
Turán's problem for hypergraphs ★★
Author(s): Turan
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ 3n $](/files/tex/f36625fb2086623f510fdcab4f53cd27a419800a.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ \frac12 n^2(5n-3) $](/files/tex/77fae1e209cf2ef781abf9d88573e4981aa00b15.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ 2n $](/files/tex/56259815f2fdf87e92dd22e0058206e8e20fb986.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ n^2(n-1) $](/files/tex/edaf55f28dcd60182c7c9d88572ea01709769bde.png)
Keywords:
4-connected graphs are not uniquely hamiltonian ★★
Author(s): Fleischner
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
Keywords:
Every prism over a 3-connected planar graph is hamiltonian. ★★
Author(s): Kaiser; Král; Rosenfeld; Ryjácek; Voss
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ G\square K_2 $](/files/tex/eb7d7828977b0484fd118a68143d72f9c6e865f3.png)
Keywords:
Hoàng-Reed Conjecture ★★★
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ C_1, \ldots, C_k $](/files/tex/0df670e43d33838d6e04e86a590da56100880e60.png)
![$ C_j $](/files/tex/b1365660549601f059d1b19f13f120a8fd821c25.png)
![$ \cup_{i=1}^{j-1}C_i $](/files/tex/b45609a301fcc110ef904d9f320f045d3947da71.png)
![$ 2 \leq j \leq k $](/files/tex/f4a8b4068d7bbb76eee5a6457f5fa87ff65184f1.png)
Keywords:
Edge-disjoint Hamilton cycles in highly strongly connected tournaments. ★★
Author(s): Thomassen
![$ k\geq 2 $](/files/tex/13bc863dca3c6b96ebfd2de373f0fe820c58b62b.png)
![$ f(k) $](/files/tex/e055b3867e7cb3cc4b2f50739eedda7657999214.png)
![$ f(k) $](/files/tex/e055b3867e7cb3cc4b2f50739eedda7657999214.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
Keywords:
Hamilton cycle in small d-diregular graphs ★★
Author(s): Jackson
An directed graph is -diregular if every vertex has indegree and outdegree at least
.
![$ d >2 $](/files/tex/6b29cd29e21416ff46f4898a33bcc7355436ee8f.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ 4d+1 $](/files/tex/5d467147b18bddde059e7178ebf5bcf6c4b4f516.png)
Keywords: