Many years ago, Grinberg found some uniquely-hamiltonian $3$-connected graphs, and published his results in a paper that has been cited several times as follows.



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> E. Grinberg, Three-connected graphs with exactly one Hamiltonian cycle, Republican Foundation of Algorithms and Programmes, Computing centre. P. Stutschka University, Riga, U.S.S.R. (1986) [in Russian].
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However the reference cannot be found (by Google or MathScinet) under this name, so I am guessing that some initial Russian-speaking author translated the details, and subsequent authors have either concurred or taken it on trust.


Of course what I really want is the description of the graphs, but I see no way of doing this without the assistance of someone able to navigate the Russian literature.


Any assistance would be gratefully received.



Grinberg's uniquely hamiltonian 3-connected graphs (Russian paper)
I have now resolved most of the mysteries, and as MO prompts me to answer my own question, I am now doing so even though it feels a bit odd.


After some false starts with expired email addresses, I managed to contact Dainis Zeps in Latvia, who kindly filled in the missing details.


Basically Zeps and Grinberg were working on unique hamiltonicity, but with Grinberg as the senior of the two. After Grinberg died in 1982, Zeps took some of Grinberg's old notes, rescued a construction for uniquely hamiltonian three-connected graphs, wrote it up, and published it under Grinberg's name. 


The construction is relatively straightforward (at least after you have seen it). If a graph has a triple of vertices $(x,y,z)$ that satisfy some simple hamiltonicity properties, then two nice things happen


* You can take two copies of this graph, connect the special triples in a particular way, and get a 3-connected uniquely hamiltonian graph, and
* The graph you have just built has lots more special triples, and so you can take two of *those* graphs and so on.


They then supplied an initial graph with the right properties. To absolutely no-one's surprise, this initial graph is based on the Petersen graph.


Zeps then did some more work on this, including some computations, and now that he felt that he had made enough of an additional contribution, wrote the viXra article that KConrad pointed out to me, now adding himself to the list of authors.


Obviously this construction produces graphs with a (cyclic) 3-edge-cut and I think the "special triple" property is essentially exactly what you would need to make a graph with a cyclic 3-edge-cut uniquely hamiltonian.


What is interesting is that the smallest of the graphs found by Grinberg/Zeps is just ONE EDGE bigger than the actual computer-determined minimum number of vertices and edges. Doing this by hand, with no computer assistance, and getting so close to the absolute smallest is pretty impressive.


Of course this means that the actual smallest graph has no cyclic 3-edge-cut, and indeed it is cyclically 4-edge-connected. It does have a cyclic 4-edge cut and so by looking at the 4 end-points $\{a,b,c,d\}$ we can come up some conditions to make this a "special quadruple". But now we just seem to be heading down a rabbit-hole constructing more and more examples that are not genuinely different.


