Geodesic
4-edge-connected 4-regular maps on the projective plane
Ars Mathematica Contemporanea, Tome 18 (2020) no. 1, pp. 51-71.

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In this paper rooted (near-)4-regular maps on the projective plane are investigated with respect to the root-valency, the number of edges, the number of inner faces, the number of nonroot-vertex-loops and the number of separating cycles. In particular, 4-edge connected 4-regular maps (which are related to the 3-flow conjecture by Tutte) are handled. Formulae of several types of rooted 4-edge-connected 4-regular maps on the projective plane are presented. Several known results on the number of 4-regular maps on the projective plane are also derived. Finally, using Darboux’s method, a nice asymptotic formula for the numbers of this type of maps is given which implies that almost every (loopless) 4-regular map on the projective plane has a separating cycle.
DOI : 10.26493/1855-3974.1615.979
Keywords: (Rooted) near-4-regular map, separating cycle, Lagrangian inversion, enumerating function, asymptotic 
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Shude Long; Han Ren. 4-edge-connected 4-regular maps on the projective plane. Ars Mathematica Contemporanea, Tome 18 (2020) no. 1, pp. 51-71. doi : 10.26493/1855-3974.1615.979. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1615.979/

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