[1] The boundary segments of the regions of Tietze's subdivision (including the segments along the boundary of the Möbius strip itself) form an embedding of Tietze's graph.

If one cuts a hole from this subdivision of the projective plane, surrounding a single vertex, the surrounded vertex is replaced by a triangle of region boundaries around the hole, giving the previously described construction of the Tietze graph.

Equivalently, the edges of Tietze's graph can be partitioned into four matchings, but no fewer.

Nevertheless, it is isomorphic to the graph J3, part of an infinite family of flower snarks introduced by R. Isaacs in 1975.

This property plays a key role in a proof that testing whether a graph can be covered by four perfect matchings is NP-complete.