Characterizations of bipartite and Eulerian partial duals of orientable hypermaps

We first rewrite the Chmutov and Vignes-Tourneret’s three-permutation formula as an explicit hyperedge-partial-duality formula in the two-permutation model, and show that in this model partial duality acts exactly by preserving the support and length of every hyperedge while reversing the $\alpha$-cycles corresponding to the selected hyperedges. Next, using the Cori and Hetyei’s construction of the medial map, we define for each hyperedge subset $E^{\mathrm{\prime }}\subseteq E(H)$ a black/white smoothing state $S_{E^{\mathrm{\prime }}}$, and prove rigorously that the state circles of $S_{E^{\mathrm{\prime }}}$ are in bijection with the vertices of the partial dual $H^{E^{\mathrm{\prime }}}$. Consequently, $H^{E^{\mathrm{\prime }}}$ is Eulerian if and only if every state circle has even length. On this basis we prove the following two main theorems: $$\begin{array}{rl}{\displaystyle H^{E^{\mathrm{\prime }}}\text{ is Eulerian}}& {\displaystyle ⟺\mathrm{\exists }\text{ a crossing-total direction }\mathrm{\Omega }\text{ of }M(H)}\\ {\displaystyle }& {\displaystyle \text{such that }{E}^{\mathrm{\prime }}=D(\mathrm{\Omega })\cup T^{\mathrm{\prime }},T^{\mathrm{\prime }}\subseteq T(\mathrm{\Omega }),}\end{array}$$ $$\begin{array}{rl}{\displaystyle H^{E^{\mathrm{\prime }}}\text{ is bipartite}}& {\displaystyle ⟺\mathrm{\exists }\text{ an all-crossing direction }\mathrm{\Phi }\text{ of }M(H)}\\ {\displaystyle }& {\displaystyle \text{such that }{E}^{\mathrm{\prime }}=C(\mathrm{\Phi })\mathrm{.}}\end{array}$$ Here $D(\mathrm{\Omega })$, $T(\mathrm{\Omega })$ and $C(\mathrm{\Phi })$ denote, respectively, the sets of all $d$-type, $t$-type and $c$-type hyperedges. Unlike the ribbon-graph case, the hypermap setting exhibits a genuine new obstruction: if some hyperedge-partial dual is bipartite, then every hyperedge of the original hypermap must have even length.