added hypergraphs note

_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 re-execution of notebooks on each build.

_toc.yml

@@ -6,4 +6,4 @@ root: intro
 chapters:
 - file: notes/o6c4c
 - file: notes/s2_nz5_flow_counterexample
-
+- file: notes/hypergraphs

notes/hypergraphs.md

@@ -0,0 +1,12 @@
+# Hypergraphs
+
+In this note we will try to find analogues of various constructions in graph theory, related to the field of cycle double covers, specifically:
+- What kind of hypergraphs should we consider?
+- What is a bridge?
+- What is a 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?

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
@@ -17,13 +25,15 @@ 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?
         - e. g., p-adic numbers (just a wild idea, not serious)
         - 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
 

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

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

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

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
+