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Bender-Canfield showed that a plethora of graph counting problems inoriented/unoriented surfaces involve two constants $t_g$ and $p_g$ for theoriented and the unoriented case respectively. T.T.Q. Le and the authorsrecently discovered a hidden relation between the sequence $t_g$ and a formalpower series solution $u(z)$ of the Painlev\'e I equation which, among otherthings, allows to give exact asymptotic expansion of $t_g$ to all orders in$1/g$ for large $g$. The paper introduces a formal power series solution $v(z)$of a Riccati equation, gives a nonlinear recursion for its coefficients and anexact asymptotic expansion to all orders in $g$ for large $g$, using the theoryof Borel transforms. In addition, we conjecture a precise relation between thesequence $p_g$ and $v(z)$. Our conjecture is motivated by the enumerativeaspects of a quartic matrix model for real symmetric matrices, and the analyticproperties of its double scaling limit. In particular, the matrix modelprovides a computation of the number of rooted quadrangulations in the2-dimensional projective plane. Our conjecture implies analyticity of the$\mathrm{O}(N)$ and $\mathrm{Sp}(N)$-types of free energy of an arbitraryclosed 3-manifold in a neighborhood of zero. Finally, we give a matrix modelcalculation of the Stokes constants, pose several problems that can be answeredby the Riemann-Hilbert approach, and provide ample numerical evidence for ourresults.

Universality and asymptotics of graph counting problems in nonorientable surfaces