It is a wellknown fact, that Moebius Ladder Graphs have $2n$ vertices, but nowhere could I find any hint of how to generalize them to Graphs with $2n+1$ vertices.

Last week I had the idea of giving up the restriction to cubic graphs and arrived at glueing together the two ends of triangle strips with $2n+1$ triangles in the "Moebius manner", i.e. with a twist.

The result is

- a $4$-regular graph
~~with exactly two edge-disjoint Hamiltonian cycles if $2n+1\ge 7$ , which contrasts the situation of Moebius Ladders, where the number of Hamiltonian cycles is different for each size and given by A124356 - OEIS and none contains a pair of edge-disjoint Hamiltionian cycles~~.- the chromatic number is $5$ for $5$ vertices and $4$ for all other cases of $2k+1$ vertices, again contrasting the situation of Moebius Ladders, where it is $2$ for $4k+2$ vertices and $3$ for $4k$ vertices (for 4 vertices it would also be $4$, but $K_4$ is normally not considered to be a Moebius Ladder)

Question:Have those Moebius Stairway graphs been described or studied already, i.e. are further special properties known?

As a remark let me explain the name "Moebius Stairway" graph: if the triangles are chosen to be isosceles right triangles and the strip is then drawn in an ascending $45^{\circ}$ angle, it looks somewhat similar to a stairway and, besides that, I liked the idea of providing an alternative to ladders.