co.combinatorics – Strong chromatic index of some cubic graphs

Definition (a bit informal) A strong edge $ k coloring of a cubic chart (3-regular) is a good $ k coloring of its edges so that any edge as well as the four adjacent edges are colored with 5 colors. the high chromatic index $ chi_S (G) $ of a cubic graph $ G $ is the smallest number $ k such as $ G $ has a strong advantage $ k coloring.

Andersen, in [1], showed that if $ G $ is a sub cubic graph (a graph of degree max 3), then $ chi_S (G) the $ 10. In the same document, he proposed the following:

Conjecture [Andersen, 1992] There is a constant $ g $ such as if a cubic graph $ G $ is such that the circumference $ gamma (G) ge g $then $ chi_S (G) = $ 5.

This conjecture is very significant, because the truth of this one would imply the truth of several notorious theoretical conjectures on graphs for all (cubic) graphs of sufficiently large circumference.

As background information on our question, some computer investigations (still very incomplete) would seem to indicate that if $ G $ has no bridge (and circumference at least 4, although we are not sure that this is really necessary), $ chi_S (G) the $ 8, and also, if $ gamma (G) ge $ 5 then $ chi_S (G) the $ 7, and if $ gamma (G) $ 9 $ then $ chi_S (G) the $ 6. Finally, we checked that the cage (3,17) indicated in [2] has no strong color edge 5, and that the (3,18) -cage in [2] has a strong edge 5 staining. We are currently trying to establish the strong chromatic index of several graphs of circumference greater than 9 listed in [2]and we are also trying to establish if the (3,19) -cage in [2] has a strong edge 5 staining. And we should probably look at a lot more graphics with a small circumference. We will update this article as this information is further verified or refuted. Our calculations are currently oriented to graphs with a circumference greater than 4. We need to check much more graphs with circumference 3 and 4 and we recognize that lack. The available computing time is limited … However, we think that our main question (question 1) is based on a solid foundation.

Before asking our question, we need a definition.

Definition Leave a $ n $-prismatic graph either a cubic graph obtained by joining two disjoint circuits of order $ n $ with a perfect match.

Our first question then is:

question 1 [main question] Let $ G $ to be prismatic. Then, the strong chromatic index of $ G $ is a maximum of 8. In addition, if the circumference of $ G $ is greater than 4, so the strong chromatic index of $ G $ As at most in our messages, prove to provide a counterexample.

The nature of a possible proof of this is almost necessarily algorithmic. An inductive proof of the more general assertion that non-bridging graphs of circumference at least 4 have a strong chromatic index at most 8 seems a little out of reach for the moment. Indeed, by working with the general graphs of circumference 4 and finding the subgraphs that are strong critics, we have found more than a thousand that are not isomorphic and large enough. Of course, some are small and occur much more often.

In [1] a linear time algorithm for finding a strong stain with at most 10 colors is given. We would also like to know if:

question 2 Is there a fast algorithm (linear time?) And simple to find a strong coloration of the outline 8 of a cubic graph without bridge?

[1] Andersen, L.D., The strong chromatic index of a cubic graph is of at most 10, Discrete mathematics, 108 (1992) 231-252

[2] Royle, G. Cubic Cages, staffhome.ecm.uwa.edu.au/~00013890/remote/cages/index.html#data

digital integration – Cubic Anharmonic Oscillator with Numerov Method

I've tried to solve the Schrödinger equation for quantum cubic anharmonic oscillator with the Numerov algorithm, but I have doubts about the probability of density (3D graph of position and time) for a consistent state. first example.

M = 20; Lamb = 0.5;
V[s_] : = 0.5 * s ^ 2 + s * [Lambda]; EM = 350.
rturn = FindRoot[V[s] == [Epsilon]m, {s, EM}][[1,2]]; d = 1 / Sqrt[2*EM]; n = round[2 (rturn/d + 4 Pi)]; s = table[-((d (n + 1))/2) + d i, {i, n}];

a[n_, d_]: = DiagonalMatrix[1 + 0 Range[n-Abs[d]], re

;
B = (a[n,-1] + 10 a[n,0] + one[n,1]) / 12;
A = (a[n,-1] - 2 a[n,0] + one[n,1]) / d ^ 2;

KE = - (1/2) * Inverse[B].A;

H = KE + Diagonal Matrix[V[s]];
{eval, evec} = Electronic system[H];
in = Order[eval];
eval = eval[[in]]; evec = evec[[in]];
evalnum = Select[eval, # <= EM &];
L = Length[evalnum];
evecnum[n_] : = evec[[n + 1]]/ Sqrt[d];
cnum = Table[Total[Conjugate[evecnum[n]]* (Pi) ^ (- 1/4) / Sqrt[(2^M)*Factorial[M]]Exp[-(s^2/2)] HermiteH[M, s]*re], {n, 0, L-1}];

realpsinum[t_] : =
Total[ Exp[-I*evalnum*t]* cnum *
Table[evecnum[n], {n, 0, L-1}]]F = 1.2;
B = ListLinePlot[Transpose[{s, Abs[realpsinum[F]]^ 2}]]

Theory of complexity – Variation of cubic space reduction of PSPACE-COMPLETE (Theoretical, complicated)

I asked myself:

if we change the definition of a PSPACE-COMPLETE definition as follows:
A B-language will be called PSPACE-COMPLETE if:

  • for each language A in PSPACE: $ A leq _ {CS} B $
  • language B belongs to PSPACE

Are there PSPACE-COMPLETE languages ​​that meet this definition?

(note: if a language A can be a cubic space brought back to the B language, which means that if there exists a reduction application from A to B that can be computed in a cubic space, we will note $ A leq _ {CS} B $, that is, if there is a determination machine M, with a band, that by entering a word w, it uses $ O (| w | ^ 3) $ space and after finishing on his tape, write the word f (w)).

it's quite confusing for me, because a pspace-complete is a problem that can be solved by using a polynomial space with respect to the input, and that every problem that can be solved in a polynomial space is mapped ( transformed by him, but I'm really not sure because of the $ O (| w | ^ 3) $ requirement.

would really appreciate your opinion about it

Perfect Matches and Cuts in Cubic Charts – Part 1

Let $ G $ to be a cubic graph without a bridge (simple) and let $ M $ to be a perfect match in $ G $. $ G-M $ will necessarily be a set of circuits. For example, if we remove a perfect match from $ K_ {3,3} $ we end up with a circuit. If we remove a perfect match from the Petersen graph, we end up with two circuits. And in general, we could have several. The next question comes to mind.

Question Let $ G $ be a cubic graph without bridge (simple) that is cyclically connected to the periphery, and let $ M $ to be a perfect match in $ G $. Is there a subset K $ edges in $ M $ such as $ G-K $ exactly two components, so that neither of the two components is a circuit, or both are circuits?

For example, any perfect match in the Petersen chart is an example of such an edge cut.

A graph $ G $ is cyclically connected to the edge 5 if no set of less than 5 edges is a separate cycle. A set K $ edges is separation cycle if $ G-K $ is disconnected and at least two of its components contain circuits.

Reverse an application of `sym` in ua` and` isoToEquiv` in cubic type theory

I prove a kind of structural invariance principle for magmas in cubic type theory with the Agda / Cubical library. This is done by building a path between two simple magmas and then carrying evidence of simple properties about that path. I have already obtained the essential of the proof (see my code repository) but I have not yet managed to complete the following lemma.

At some point in the evidence, I have the following data:

  • A bijection between types: f: ℕ →
  • The isomorphism built with F: More: Iso ℕ ℕ₀
  • A function that gives the isomorphism inverse of an isomorphism: invIso: Iso A B → Iso B A

Now, I would like to prove that:

sym (ua (isoToEquiv fIso)) ua (isoToEquiv (invIso fIso also))

My question has two parts:

  • Is this a valid theorem in HoTT? Although this statement seems valid, I may have made a false statement?
  • Are there built-in functions in Agda / Cubical that can help the proof?

co.combinatorics – Colorability of edges and Hamiltonicity of a certain class of cubic graphs

Let $ G $ to be a simple cubic graph (that is to say 3-regular). A { it dominant circuit} of $ G $ is a circuit $ C $ so that each edge of $ G $ has an endvertex in $ C $. The circuit $ C $ is { it chordless} if no edge that is not in $ C $ has both endvertices in $ C $ (These edges are called { agreements it} $ C $). An { it MO graph} is a simple cubic graph with a dominant circuit without chords.

QUESTION 1: Are MO graphs edge-3-colorable?

QUESTION 2: Are MO graphs Hamiltonian? An affirmative answer to this question implies an affirmative answer to the first.

Computer-generated experiments with randomly generated MO graphs, with fairly large graphs and thousands, suggest that the answer to these two questions is YES. I hope that at least question 1 can be settled. Question 2 seems difficult.

Non-decimal representation of the solutions of the cubic equation

I'm trying to solve a cubic equation as follows

Solve[2 (Sqrt[x]) ^ 3 - 1.5 * x ^ 0.5 * vd == 2 (Sqrt)[[Eta]0]) ^ 3, {x}

I want to get the solution from $ x $ in terms of $ eta_0 $, with rational / irrational factors, then in the limit $ vd << eta_0, x $ get the solution.
When I try to use solve, I get a solution like this

enter the description of the image here
Since I am trying to solve an analytic physical equation, I want to get the good prerequisites in terms of real numbers (rational or irrational) and not their decimal values ​​(like $ sqrt {2} $ instead of 1.414 and $ frac {3} {5} $ instead of 0.6 in pre factors.

Theory of cubic types – What does "Kan" mean in "Kan Operations"?

I am studying the Cubic Type Theory and I see the word "Kan Operations" (ref1, ref2, ref3, and there are many), which is related to "adding a cap." to a tube "(can also be explained using" given path between a and band two paths between a and c/b and re respectively we can use Kan operations to get a path between c and re").

I wonder if there are references that can explain the origin of the word "Kan" – what does it mean? Are the "Kan extensions" in category theory?

cubic equations – how to find all primes p for which there is a positive integer n such that p ^ n + 1 is the square of a positive integer?

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curves – Weighting of a cubic spline of hermit

I'm trying to understand a function behind the software curve drawing algorithm. Originally, each node comes with 3 parameters: hour, value and tangent. I found that this corresponds to Hermite's cubic spline and I confirmed that the use of the Wikipedia equation gave me the same result as the evaluation of the software.

The next curve is when both tangents are 0.

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The next curve is when both tangents are 1 (45 degrees).

enter the description of the image here

But this software has a value of "weight" on each node, which ranges from 0 to 1. I'm trying to understand how the weight could affect the spline empirically. Here is my observation:

The weight varies from 0 to 1. In this gif below, slide the handle to the right to increase the weight. Drag upward stretching does not affect the weight (regardless of tangent modification)

enter the description of the image here

When the two tangents are 0, with applied weight and the two weights are 0.3333333, resulting in the same form as if no weight is applied. (Smooth S-shape, first image) I think it's the biggest clue, 0.3333333 may have something to do with the "cubic" function. Below an image of this.

enter the description of the image here

When the two tangents are equal to 0, with the applied weight and the two weights equal to 1 at the most, the curve is inclined further in the X axis to join at the center between 2 points. Indicate this, weight 1 does not mean unweighted as a weight function would behave, but rather really the maximum weight possible. Below an image of this.

enter the description of the image here

When the two tangents are equal to 0, with the weight applied and the two weights equal to 0, give a linear graph as if their tangents were equal to 1. Below, an image of that.

enter the description of the image here

When the two tangents are equal to 1, the weight does not affect their shape, no matter the value. It remains linear. Suggest that the weight affects the components that have become tangent, but canceled when the two components are equal. I guess it did something to the cos component? Since, with the maximum weight, the graph is tilted further down the X axis.

However, if the other tangent is not 1, changing the weight of the side with tangent 1 affects the shape. Indicate that weight is not simply "weight the tangent on this side".

If anyone can determine where the weight should be applied (or any spline example you know with a weight setting), this is highly appreciated. Thank you.