Edge colorability of small d/k graphs - among the largest known graphs for the undirected degree diameter problem What is known about the edge colorability of the graphs residing in the small $d/k$ section in this table (upper left corner) ?
For example, what is the chromatic index of the $d=4$, $k=4$ graph with $41$ vertices in that table found by James Allwright? Is it $4$ or $5$ ?
Is there any research on this topic ?
Could you point to relevant papers on this topic ?
 A: I checked the 41-vertex d=4 k=3 graph by James Allwright and found that it has a four clique in its line graph. 
I also checked the linegraph with this program  but after a day of computing it did not find a 4 coloring.
After reading this paper I learned that this problem (whether the James graph is 4 or 5 edge colorable) can be nowadays solved exactly in a few hours on moder desktop machine having a few GFLOPs. There is an O(1.82^n) algorithm to do that, as it is referenced in the paper.
After reading some more, I "discovered" that it is known that every regular odd order graph is class two. So the James graph is class two as well. 
I checked the linegraph of Exoo's 72 graph with smallk as well and it could be five colored. 
Similarly, d5k2n24's line graph can be 5 colored as well, using the smallk program, it takes a few miliseconds. 
As a comparison, for the line graph of the Hoffman Singleton graph the smallk program did not find a 7 coloring after a few minutes of computing even though such 7 coloring is known to exist.
Also, Exoo's n=98 d=4 k=4 graph can be 4 edge colored, according to a few milliseconds of computation using the smallk program on the line-graph of Exoo's n=98 d=4 k=4 graph.
I found a 3 coloring using the smallk program on the line graph of d3k3n20 as well after a few milliseconds of computation.
So to summarise the story (ci=chromatic index) :
d=3 k=2 n=10 ci=4, class 2
d=3 k=3 n=20 ci=3, class 1
d=4 k=2 n=15 ci=5  class 2
d=4 k=4 n=41 ci=5  class 2
d=4 k=4 n=98 ci=4, class 1
d=5 k=2 n=24 ci=5, class 1
d=5 k=3 n=72 ci=5, class 1
d=7 k=2 n=50 ci=7, class 1

