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I would like to know whether there are known constructions which provide a bijection between loops (isomorphism classes) and (possibly directed) graphs. Any reference to a useful paper in this direction will be appreciated.

Searching combinations of words "graphs, loops, representation, bijection" the hits contain too many irrelevant links, so I get staked. Thank you in advance for any help.

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It isn't hard to define a class of graphs whose isomorphism classes correspond to the isomorphism classes of loops. It is easiest using vertex colours: see B. D. McKay, A. Meynert and W. Myrvold, Small Latin squares, quasigroups and loops, J. Combinatorial Designs, 15 (2007) 98-119.

In this corrected version the construction is $G_3(L)$ on page 13. You might have to adjust it a little to achieve exactly what you need.

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  • $\begingroup$ Thank you very much. This knowledge is very helpful. btw. Is there known any available database of isomorphism classes of loops of order 7? From the LOOPS package of GAP I can get them systematically up to order 6. For higher orders it can generate a loop randomly, but not systematically. $\endgroup$
    – Stefan Gyürki
    Commented Oct 20, 2014 at 11:21
  • $\begingroup$ @Stefan: I'm not aware of such a list, though it would be routine to make them as there are only 23,746. If you can't find a list soon (or someone tells us where one is) let me know and I'll make them. $\endgroup$
    – Brendan McKay
    Commented Oct 20, 2014 at 13:57
  • $\begingroup$ I have checked homepages of Andries Brouwer, Patric Ostergard, Ted Spence, Ian Wanless and yours (Did I miss anybody?) but no such data been found. $\endgroup$
    – Stefan Gyürki
    Commented Oct 20, 2014 at 20:44
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    $\begingroup$ They are now on my Latin Squares page: cs.anu.edu.au/~bdm/data/latin.html $\endgroup$
    – Brendan McKay
    Commented Oct 21, 2014 at 13:59
  • $\begingroup$ I think we have such a construction without using colours. Are you aware of such result without colours? I couldn't find any reference neither in your paper, nor elsewhere. $\endgroup$
    – Stefan Gyürki
    Commented May 23, 2015 at 12:43

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