Strong chromatic index of some cubic graphs


Definition (somewhat informal) A strong edge $ k$ coloring of a cubic (3-regular) graph is a proper $ k$ coloring of its edges so that any edge together the four edges adjacent to it are colored with 5 colors. The strong chromatic index $ \chi_S(G)$ of a cubic graph $ G$ is the smallest number $ k$ such that $ G$ has a strong edge $ k$ coloring.

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

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

This conjecture is highly significant, as the truth of it would imply the truth of several notorious graph theoretical conjectures for all (cubic) graphs of large enough girth.

As a bit of background information to our question ahead, some (as yet very incomplete) computer investigations would seem to indicate that if $ G$ is bridgeless (and of girth at least 4, although we are not sure this is really needed), then $ \chi_S(G)\le 8$ , and also, if $ \gamma(G)\ge 5$ then $ \chi_S(G)\le 7$ , and if $ \gamma(G)\ge 9$ then $ \chi_S(G)\le 6$ . Finally, we have verified that the (3,17)-cage listed in [2] does not have a strong edge 5 coloring, and that the (3,18)-cage in [2] has a strong edge 5 coloring. We are currently trying to establish the strong chromatic index of several graphs of girth larger than 9 listed in [2], and we are also trying to establish whether the (3,19)-cage in [2] has a strong edge 5 coloring. And we should probably look at a lot more graphs with small girth. We will update this post as this information is further verified, or refuted – our computations are geared at the moment towards graphs with girth higher than 4. We need to do a lot more checking on graphs with girth 3 and 4 and we acknowledge that this is lacking. There is only so much compute time available … however, we believe there is a firm basis for our main question (question 1) ahead.

Before we pose our question, we need a definition.

Definition Let an $ n$ -prismatic graph be a cubic graph obtained by joining two disjoint circuits of order $ n$ with a perfect matching.

Our first question is then:

Question 1 [main question] Let $ G$ be prismatic. Then the strong chromatic index of $ G$ is at most 8. Moreover, if the girth of $ G$ is greater than 4, then the strong chromatic index of $ G$ is at most 7. As always in our posts, prove of provide a counterexample.

The nature of a possible proof of this is almost necessarily algorithmic. An inductive proof of the more general statement that bridgeless graphs of girth at least 4 have strong chromatic index at most 8 seems a bit out of reach at the moment. Indeed, in working with general graphs of girth 4 and finding subgraphs which are strong 8-critical, we have found over a thousand that are not isomorphic, and fairly large. Of course, there are a few that are small and that occur a lot more often.

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

Question 2 Is there a fast (linear time?) and simple algorithm to find a strong edge 8 coloring of a bridgeless cubic graph?

[1] Andersen, L. D., The strong chromatic index of a cubic graph is 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

Finite index subgroup of $GL_n(\Bbb C)$ and Chevalley groups

I’m trying to show that if $ G$ is a Chevalley group, then every finite indexed subgroup of $ G(\Bbb Z)$ is Zariski dense in $ G(\Bbb C)$ . ($ G(\Bbb Z)$ is the Chevalley group over $ \Bbb Z$ and similarly for $ G(\Bbb C)$ )
But I’m struggling to understand some basic stuff, It seems to me that there can’t be finite index subgroup in $ G(\Bbb C)$ how can I show that?
also is there a good source that covers Chevalley groups over $ \Bbb Z$ ?

Find smallest index that is identical to the value in an array

The task is taken from LeetCode

Given an array A of distinct integers sorted in ascending order, return the smallest index i that satisfies A[i] == i. Return -1 if no such i exists.

Example 1:

Input: [-10,-5,0,3,7] Output: 3 Explanation:  // For the given array, A[0] = -10, A[1] = -5, A[2] = 0, A[3] = 3, thus the output is 3. 

Example 2:

Input: [0,2,5,8,17] Output: 0 Explanation:  // A[0] = 0, thus the output is 0. 

Example 3:

Input: [-10,-5,3,4,7,9] Output: -1 Explanation:  // There is no such i that A[i] = i, thus the output is -1. 

Note:

1 <= A.length < 10^4

-10^9 <= A[i] <= 10^9

My solution

The solution has time complexity of $ O(n)$ and space complexity of $ O(1)$ . I start to look from the start to the last element. If I find a value that is greater than i, then I can exit early (because there won’t be an element that is equal to i anymore). If I find A[i] === i, then I have a result.

Is there a faster solution than the one provided?

/**  * @param {number[]} A  * @return {number}  */ var fixedPoint = function(A) {     for (let i = 0; i < A.length; i++) {         if (A[i] > i) { return -1; }         if (A[i] === i) { return i; }     }     return -1; }; 

Updating package index is very slow due to large ‘Contents’ entries

When updating my package index, e.g. over apt update it always has to download some very large entries, usually 20 – 40 MB with a Contents-* prefix. This slows down the update process very much also due to the reason that it seems like these packages are fetched multiple times. Usually they are named something like Contents-i386 or Contents-amd64.

I’m absolutely not sure if this is normal, but I’m a Ubuntu user for quiet some years and can’t remember, that index updates where that heavy in the past. The problem could be related to some configuration change I had to made in the past but I can’t really remember what it was.

A uniform upper bound for Fredholm index of quasi Laplace operators on a compact parallelizable manifold

Assume that $ M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $ D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$ where $ \{X_1,X_2,\ldots,X_n\}$ is a global smooth frame?

magento 2.3.1 elastic search index name

I have recently updated our store to magento 2.3.1 it seems to work fine but when we run it with Elastic search we get error:

fielddata is disabled on text fields by default

I enabled that using a curl call but every time the indexer runs it creates a new index with a new name i.e magento2_product_1_v{index/count}. The version number at the end of the index name changes everything. So every time indexer runs I enabled it manually via curl. I was wondering if there is a way to permanently enable fielddata?

thanks

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Strong chromatic index of cubic gaphs of very high girth


Definition (informal) Let $ G$ be a cubic graph (no other conditions are placed). A strong edge coloring of $ G$ is a proper coloring of the edges with at least 5 colors such that for every edge $ e\in E(G)$ , five different colors are used to color $ e$ and its four adjacent edges. The strong chromatic index $ \chi_S(G)$ of $ G$ is the smallest number of colors required to get a strong edge coloring of $ G$ .

There are a number of open conjectures related to the strong chromatic index of cubic graphs. One such conjecture is the following:

Conjecture [Andersen?,1992,[1]] There is a constant $ k$ , such that for graphs of girth $ \gamma(G)\ge k$ one has $ \chi_S(G)=5$ .

We would like to investigate this conjecture. The truth of it has many implications for graphs in general.

One can see from computer experiments (not exhaustive) that already for $ \gamma(G)\ge 5$ one would seem to have $ \chi_S(G)\le 7$ . For $ \gamma(G)\ge 9$ it would appear that $ \chi_S(G)\le 6$ , although surely this requires further testing. We wonder how high the girth must be to bring down the strong chromatic index to 5. We do not have graphs of girth higher than 9 available for computer experiments. So our question is the following (note we have no notion of what cubic graph generation is like):

Question Is there a (public?) implementation of a cubic graph generating function where the girth can be made arbitrarily high? Ideally, such a function generates graphs that are not isomorphic, in some sequence, for a specified (and possible) number of vertices, and it can be started and stopped at will. The idea is to generate several examples (in the hundreds), perhaps exhaustively for relatively few vertices, in order to carry out some computer experiments that would allow us to “guess” the right value for the girth required to get a strong chromatic index equal to 5. Ideally, the graphs produced are in graph6 format. Ideally also, the function, if publicly available, is compiled and can be run from Mathematica (in general, this is possible).

Alternatively, is there a list of cubic graphs with very high girths (higher than 9, probably significantly so)?

I am relatively new to this site; if this question (request) is inappropriate for MO, leave a comment and explanation, and it will be deleted.

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

[2] Horák, P, Qing, H., Trotter, W., Induced matchings in cubic graphs, Journal of Graph Theory 17 2 (1993) 151-160

Undefined index: und on .tpl.php for content type

How can i fix the error message?

the error message is Notice: Undefined index: und in include() (line 9 of /var/www/html/sites/all/modules/custom/theme/icons.tpl.php).

my code is:

<?php  global $  language ;   global $  base_url , $  base_path ; $  lang = $  language->language; $  count = 0; foreach ($  data as $  kay => $  node_obj) {     $  node = node_load($  node_obj->nid); if (isset($  node->field_description[$  lang])) {     $  img_url = $  node->field_icon['und'][0]['uri'] ; $  count++; 

?>