Typically Cayley graphs are defined for groups and generators sets S. But basically one only needs some set S and another set V and partially defined operation SxV->V, then one defines graph with vertices V and one have an edge between $v_1,v_2$ if there exists $s \in S: v_1=sv_2$. So for definition nothing is required - neither associativity, neither identity and so definition works various types of algebraic structures: groupoids, monoids, quasigroups, etc...
The famous Lovász conjecture predicts existence of the Hamiltonian path for Cayley graphs of arbitrary finite groups.
Question 1: Are there any results or expectations when we somehow relax group condition to groupoid, loop, semigroup, or whatever...
Question 2: In particular graphs of states/transitions for famous 15-game or its NxN analogs, or other sliding puzzles or M13 Mathieu groupoid by Conway - are they expected to contain a Hamiltonian path ?
Question 3: One can also take Octonions, Sedenions, Pathions, Chingons (etc. by Cayley–Dickson construction ) take their units and construct "Cayley" graphs from these units - same question .
All graphs are considered undirected. (Any information for directed graphs is also welcome, but it is already known for groups that directed Hamiltonian paths not always exists).
