From 7330a25aff6d291b4b27b54919960eb5581e1243 Mon Sep 17 00:00:00 2001 From: nikolay-ulyanov Date: Fri, 4 Oct 2024 21:16:42 +0200 Subject: [PATCH] added hypergraphs note --- _config.yml | 4 +- _toc.yml | 2 +- logo.png | Bin 9854 -> 787762 bytes notes/hypergraphs.md | 12 +++++ stuff/txt_notes/graph_book_plan.txt | 21 ++++++--- stuff/txt_notes/graphs_important_todo.txt | 3 ++ stuff/txt_notes/graphs_todo_after_paper1.txt | 44 +++++++++++++++++-- stuff/txt_notes/hypergraphs.txt | 38 +++++++++++++++- stuff/txt_notes/paper1.txt | 10 +++++ 9 files changed, 120 insertions(+), 14 deletions(-) create mode 100644 notes/hypergraphs.md diff --git a/_config.yml b/_config.yml index 5f534f8..e043549 100644 --- a/_config.yml +++ b/_config.yml @@ -1,8 +1,8 @@ # Book settings # Learn more at https://jupyterbook.org/customize/config.html -title: My sample book -author: The Jupyter Book Community +title: Computational Graph Decompositions +author: Nikolay Ulyanov / gexahedron logo: logo.png # Force 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cycle? +- What kind of regular cover can they have? +- What is a flow and a nowhere-zero flow on hypergraphs? +- What is a perfect matching? +- Can we find a regular cover with perfect matchings? +- Can we find an analogue or orientable constructions and covers? +- What about signed analogues? diff --git a/stuff/txt_notes/graph_book_plan.txt b/stuff/txt_notes/graph_book_plan.txt index eb9a0a4..4421242 100644 --- a/stuff/txt_notes/graph_book_plan.txt +++ b/stuff/txt_notes/graph_book_plan.txt @@ -1,6 +1,14 @@ # Graph Book jupyter-book stuff: +TODOs: + - add navigation with arrows + - add favicon + - remove © Copyright 2023 + - add numeration + - change fonts + + ## TODO: How to add References https://jupyterbook.org/en/stable/content/references.html https://jupyterbook.org/en/stable/tutorials/references.html @@ -17,6 +25,7 @@ https://www.tomasbeuzen.com/python-programming-for-data-science/README.html - understand: + - 6c4c progress in last years - nz6-flows - Z6-connectivity - TODO: maybe we could try some structures, with a+b=c? @@ -24,6 +33,7 @@ https://www.tomasbeuzen.com/python-programming-for-data-science/README.html - or something with zero divisors - Clifford algebra - or something non-commutative, but associative and preserving order as in quaternions + - signed graphs - Paper2, on unit vector flows: @@ -93,7 +103,7 @@ theorems I like: Solution of the bipartite analogue of the Oberwolfach problem Road coloring Theorem Smith's Theorem about hamiltonian cycles - Snark Theorem (probably, there's still 1 paper missing for the full proof) + Snark Theorem (but there's still 1 paper missing for the full proof) Construction of cyclically 6-connected snarks @@ -102,7 +112,7 @@ theorems I like: https://www.youtube.com/watch?v=rXw1Fb93vlQ 1. resolving quartic, 10:52 familiar formulas - same as in nz4 flow <=> orientable double 4-cover (theorem 13.1.7) + same as in nz4 flow <=> orientable 4-cycle double cover (theorem 13.1.7) https://en.wikipedia.org/wiki/Quartic_equation#Galois_theory_and_factorization 2. Schläfli graph, The intersection graph of the 27 lines on a cubic surface is a locally linear graph that is the complement of the Schläfli graph can we find analogue for Petersen graph? I believe we can, @@ -117,7 +127,6 @@ theorems I like: TODO: or dynamically change edge order, based on bruteforce stats - render book - Work on the paper1 -- build S2 uvf for 18.05g2 - what is the minimal possible rich-connected-component (judging by edge count) - for Petersen colouring - for 6c4c @@ -214,8 +223,8 @@ theorems I like: + - Graceful labeling, beta+seq - graham-haggkvist -- EPPDC - Tree packing, matrix packing +- EPPDC (good heuristic code, but also on the border of a failure) - P1f stuff (mostly failures here) TODO: or=0 - https://gexahedroet.livejournal.com/18152.html @@ -360,7 +369,8 @@ TODO: youtube - https://www.youtube.com/watch?v=iFNaulZV10k -TODO: does having Petersen coloring for snarks imply in some sense having a Petersen graph minor? and if so, proving snark theorem, and if so proving 4CT? +TODO: does having Petersen coloring for snarks imply in some sense having a Petersen graph minor? + and if so, proving snark theorem, and if so proving 4CT? not CQ Zhang: - NP-completeness @@ -456,7 +466,6 @@ http://www.mat.savba.sk/~kochol/ Open ended questions/neighbour fields: - some (flawed) philosophy -- o5cdc corresponds to some gluing of faces; can we do something similar for o6c4c? - tropical geometry; hodge theory of matroids; chip-firing on graphs and Riemann-Roch theory - some other research stuff I found with googling, that is related to some keywords diff --git a/stuff/txt_notes/graphs_important_todo.txt b/stuff/txt_notes/graphs_important_todo.txt index fa44718..c981c15 100644 --- a/stuff/txt_notes/graphs_important_todo.txt +++ b/stuff/txt_notes/graphs_important_todo.txt @@ -41,6 +41,9 @@ main TODO: ==== +- ABELIAN TROPICAL COVERS + mentions and reformulates nowhere-zero 5-flow + todo, поток мыслей, но что-то не склеивается: nowhere zero flows diff --git a/stuff/txt_notes/graphs_todo_after_paper1.txt b/stuff/txt_notes/graphs_todo_after_paper1.txt index 91e4bf7..d20f439 100644 --- a/stuff/txt_notes/graphs_todo_after_paper1.txt +++ b/stuff/txt_notes/graphs_todo_after_paper1.txt @@ -1,12 +1,35 @@ +TODO: find more unit vector flows + e. g. for all 20.05 graphs + - DONE: also check the third 18.05cyc3 graph - is it interesting? + (base) ➜ snarkhunter-2.0 git:(main) ✗ ./snarkhunter 18 5 S s C3 + 32 bit mode + ./snarkhunter 18 5 S s C2 + All done, 3 graphs generated with 18 vertices. + CPU time: 0.0 seconds. + it's because C3, not C4 + so it's "cyclically 3-edge-connected" + - ok, I checked it, looks like it's derivative of Petersen graph + and having cyc4 is important for non-trivialness of uvf-flow + - I checked one of uvf-o6c4c flows, + there 3-cut edges are all poor, all other edges are rich, + so I guess it's a gluing of 2 Petersen graphs, with a 3-cut + +TODO: Fano colorings of o6c4c solutions, are they interesting? + +TODO: colouring defect 3, optimal 3-arrays + DONE! or=0 produces o6cdc -DONE: or=0; rich pm edges from same layer, how are they distributed in o6cdc - in each layer we get even number of them; maybe even on each circuit; maybe it's even obvious how we prove it, actually, let's just follow the circuits and see what rich edges do +DONE: or=0; rich pm edges from same layer, how are they distributed in o6cdc - + in each layer we get even number of them; maybe even on each circuit; + maybe it's even obvious how we prove it, actually, let's just follow the circuits and see what rich edges do Problem 7.2. Does there exist a nontrivial snark with defect 3 in which every core hexagon is single-core? -This problem is particularly interesting from the point of view of Fulkerson’s conjecture. If such a snark did exist, then either its Fulkerson cover would not consist of two -complementary optimal 3-arrays, or else the snark would provide a counterexample to -Fulkerson’s conjecture. +This problem is particularly interesting from the point of view of Fulkerson’s conjecture. + If such a snark did exist, then either its Fulkerson cover would not consist of two + complementary optimal 3-arrays, or else the snark would provide a counterexample to + Fulkerson’s conjecture. TODO: 12>34>56 encoding, symmetries: @@ -58,9 +81,19 @@ TODO: Concretely, the Tutte-Coxeter graph can be defined from a 4-dimensional sy https://github.com/sagemath/sage/pull/38218 +TODO: or=2 + +TODO: Fan-Raspaud triples TODO: circular flow number +TODO: + - Snarks with circular flow number 5 + - Snarks which require at least 5 pms to cover edges + + also check paper "Perfect matching index vs. circular flow number of a cubic graph" + https://arxiv.org/abs/2008.04775 + ... + TODO: petersen graph, 9+6+ribbon5x3: can we use it for 2d nz-flow? TODO: check all 3 versions, that are different from 96555 solution TODO: ribbon graphs @@ -105,6 +138,7 @@ TODO: has_2cdcs, inconsistent => or>=5? TODO: has_2cdcs, where there are no pm edges, connecting circuits from same partition - has_3cdcs + - hypergraphs TODO: circuits_odd_poor=0,2,4 vs r244odd @@ -168,3 +202,5 @@ TODO: or=0, t2unor=0: Do we extend to signed graphs? (probably not) TODO: or=0, study t2or and t2unor Todo: or=0, are pms related to 3-coloring? + +TODO: r-graphs diff --git a/stuff/txt_notes/hypergraphs.txt b/stuff/txt_notes/hypergraphs.txt index 9270db4..23ebcc0 100644 --- a/stuff/txt_notes/hypergraphs.txt +++ b/stuff/txt_notes/hypergraphs.txt @@ -2,11 +2,21 @@ todo: доказать, что 3-uniform 3-regular linear hypergraphs: - bridgeless + UPD: so this is easy then? + in a sense that we always have a nz3-flow + because this bipartite graph, + which corresponds to hypergraph + it has nz3-flow always, and it's always bridgeless + TODO: or did i mean something different by bridgeless here? - have 3 hamiltonian cycles, которые образуют balanced triple cover прям сразу hamiltonian? TODO: Richmond-Cremona configuration +todo: nz-flows + ... + + - what other generalizations possible, beyond matroids and signed graphs? maybe continuous analogs? - ? hypergraphs, (Steiner) triple systems - Fano plane (? as analog of K4? Petersen graph?) @@ -153,7 +163,8 @@ assert: m == n edge_pair_count_in_bc3c[e1][e2] total_edge_count_in_bc3c вроде бы код показывает, что всегда есть накрытия -надо теперь порисовать эти накрытия, наверняка там бага (потому что мне часто показывается, что есть накрытия в 3 слоя, на 7 вершинах) +надо теперь порисовать эти накрытия, наверняка там бага + (потому что мне часто показывается, что есть накрытия в 3 слоя, на 7 вершинах) нашлись графы на 10 вершинах без решений @@ -228,3 +239,28 @@ i wonder - на скольки вершинах я найду сложный г то по идее есть matching ребро - вершина так или нет? + + + +cubic graph with bridge: + 0-1, 0-... + 0 and 1 are center vertices + so no need to describe what happens in the other 0 edges + 1-2, 1-3 + 2-4, 2-5 + 3-4, 3-5 + 4-5 + +okay, now how we convert it to a bipartite graph? + let's label partitions by oddness of vertex values + 1-2, 1-4 + 2-3, 2-5 + 4-7, 4-9 + 3-6, + + wait + every cubic graph has a Tait coloring + <=> 3-edge-coloring + => at worst nz4-flow, right? + because we have 2-colored cycles then + so we don't have bridges then diff --git a/stuff/txt_notes/paper1.txt b/stuff/txt_notes/paper1.txt index 31acce3..f3b2dd1 100644 --- a/stuff/txt_notes/paper1.txt +++ b/stuff/txt_notes/paper1.txt @@ -1350,3 +1350,13 @@ o5cdc for Petersen graph: ok, looks like there are no Beltrami fields for nz5-flows at least in this interpretation + + +or=0 => s2=3 + no idea + can we use 12>34>56 encoding to help with this? + i don't know + + we also can look at cases of or=2 + and check how 12>34>56 encoding is affected + \ No newline at end of file

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