Definition(a bit informal) Astrong edge $ k coloringof 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. thehigh 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.

DefinitionLeave a$ n $-prismaticgraph 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 2Is 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