
Rainbow Proof Shows Graphs Have Uniform Parts - c89X
https://www.quantamagazine.org/mathematicians-prove-ringels-graph-theory-conjecture-20200219/
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fyp
I read until the end and it turns out to be a nonconstructive proof via the
probabilistic method!

[https://en.wikipedia.org/wiki/Probabilistic_method](https://en.wikipedia.org/wiki/Probabilistic_method)

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maweki
I did not completely understand to what graphs this applies. Certainly not
generic ones, that can't be true.

Is it only complete ones? It seems intuitively true that complete graphs can
be tiled this way.

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Sharlin
I was wondering the same thing. It does concern complete graphs only, but note
also the additional requirement of edge-disjointedness (that is, every edge in
the large graph is covered by exactly one copy of the subgraph).

> _A typical decomposition question asks whether the edges of some graph G can
> be partitioned into disjoint copies of another graph H. One of the oldest
> and best known conjectures in this area, posed by Ringel in 1963, concerns
> the decomposition of complete graphs into edge-disjoint copies of a tree. It
> says that any tree with n edges packs 2n+1 times into the complete graph
> K2n+1. In this paper, we prove this conjecture for large n._

[https://arxiv.org/abs/2001.02665](https://arxiv.org/abs/2001.02665)

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pvaldes
Is this an open door to graph classification or simplification?

~~~
lmeyerov
Re:classification... afaict, only in the rare case of complete graphs. You can
extract those.. but they're dull. The use of rainbow coloring _IS_ indicative
of what folks do for classification & motif finding already on real-world
graphs and even tables:

\-- Spectral methods: Similar to looking at the top eigen values for PCA, they
score the connectivity matrix. I haven't seen a great explanation of this, but
the Microsoft Defender talk at BlueHat last year shows a decent place where to
use it.

\-- Counting types in weakly connected components of a property graphs. Most
graphs connect already-classified entities, and tons of methods let you get
components out of them. You can get quite far just by making a vector like
"cluster 3 has: two reds, three yellows". For a security graph, something
like, "3 different IPs, and two alerts of type X". Matt Swann's talk at last
year's [https://www.graphtheplanet.com/](https://www.graphtheplanet.com/) on a
MS internal security team's scalable approach is great here. (We're about to
release the last wave of tickets to this year's GraphThePlanet, if you're in
SF next week!).

\-- Tables: At the same Microsoft BlueHat event as the spectral methods talk,
I demoed how to understand the structure behind a log alert dump via a simple
& automatic 'hypergraph transform' that lets you use any of these methods even
on regular data tables and logs:
[https://twitter.com/Graphistry/status/1189966458844930048](https://twitter.com/Graphistry/status/1189966458844930048)
(jump to 12:45). This is how ~half of our users use Graphistry, and a
generalization of what Matt does.

There are a bunch of papers on 'motif mining', and in deep learning, handling
graph-y domains via random walks, and they often come down to a variant of #2
above. For deep learning, I'm curious about combining w/ #3 to go from 2010's
era ~perception-on-pixels -> 2020 era ~behavior-on-traces.

~~~
orange3xchicken
Just re spectral methods on graphs: There are a number of really interesting
interpretations of spectral methods including energy stabilization of spring
networks and discrete analogues to heat diffusion over grids. The area is
really fascinating.

One general interpretation is that the eigenvalues of the adjacency & other
matrices that capture the connectivity of the graph are solutions to
optimization problems [0]. Usually, these optimization problems and their
solutions correspond to relaxations of interesting combinatorial features or
operations on graphs - e.g. different ways of cutting graphs to yield binary
partitions on the nodes (spectral clustering [1]).

[0] [https://en.wikipedia.org/wiki/Min-
max_theorem](https://en.wikipedia.org/wiki/Min-max_theorem)

[1]
[https://en.wikipedia.org/wiki/Spectral_clustering](https://en.wikipedia.org/wiki/Spectral_clustering)

