**Definition** (somewhat informal) A *strong edge $ k $ Colorant* of a cubic graph (3-regular) is a suitable $ k $ Color your edges so that any edge next to the four adjacent edges is colored with 5 colors. the *strong color index* $ chi_S (G) $ of a cubic graph $ G $ It is the smallest number $ k $ such that $ G $ it has a strong advantage $ k $ Colorant.

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

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

This conjecture is highly significant, since the truth would imply the truth of several notorious graphical theoretical conjectures for all graphs (cubic) of sufficiently large circumference.

As some background information for our future question, some computer investigations (still very incomplete) seem to indicate that if $ G $ has no bridges (and at least 4 circumferences, although we are not sure that this is really necessary), then $ chi_S (G) le 8 $, and also, if $ gamma (G) ge 5 $ so $ chi_S (G) le 7 $, what if $ gamma (G) ge 9 $ so $ chi_S (G) le 6 $. Finally, we have verified that the (3,17) -box listed in [2] does not have a strong edge 5 to color, and that the (3,18) -box in [2] It has a strong edge 5 to color. We are currently trying to establish the strong color index of several girth graphs of more than 9 listed in [2], and we are also trying to establish if the (3,19) -box in [2] It has a strong edge 5 to color. And we should probably see many more graphics with a small circumference. We will update this publication as this information is verified or refuted. Our calculations are oriented at this time towards graphs with a circumference greater than 4. We need to do much more in the graphs with the circumference 3 and 4 and we recognize that this is missing. There is only so much computing time available … however, we believe there is a firm basis for our main question (question 1) ahead.

Before asking our question, we need a definition.

**Definition** Leave a *$ n $-prismatic* graph is a cubic graph obtained by joining two separate circuits of order $ n $ with a perfect combination.

Our first question is then:

**Question 1** [main question] Leave $ G $ be prismatic Then the strong chromatic index of $ G $ is at most 8. In addition, if the circumference of $ G $ is greater than 4, then the strong chromatic index of $ G $ is at most 7. As always in our messages, try to provide a counterexample.

The nature of a possible proof of this is almost necessarily algorithmic. An inductive test of the more general claim that the graphics with no girth gap at least 4 have a strong chromatic index at most 8 seems a bit out of reach at this time. In fact, when working with general graphs of circumference 4 and finding subgraphs that are strong 8-critical, we find more than a thousand that are not isomorphic, and they are quite large. Of course, there are some that are small and that happen a lot more often.

In [1] A linear time algorithm is provided to find a strong coloration with a maximum of 10 colors. We would also like to know if:

**Question 2** Is there a fast (linear time) and simple algorithm to find a strong edge coloration 8 of a cubic graph with no bridges?

[1] Andersen, L. D., The strong color 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