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\title{\bf Counting $2$-Connected $4$-Regular Maps on the Projective Plane}

\author{Shude Long\thanks{Supported by Natural Science Foundation Project of
Chongqing grant cstc2012jjA00041 and the Innovation Foundation of Chongqing grant
KJTD201321.}\\
\small Department of Mathematics\\[-0.8ex]
\small Chongqing University of
Arts and Sciences,\\[-0.8ex]
\small  Chongqing  402160, P.R.China\\
\small\tt longshude@163.com\\
\and
Han Ren \thanks{Supported by the NNSFC grant 11171114. The corresponding author.}\\
\small Department of Mathematics\\[-0.8ex]
\small East China Normal University,\\[-0.8ex]
\small Shanghai 200062, P.R.China\\
\small\tt hren@math.ecnu.edu.cn}
\date{\dateline{Jan 1, 2012} {Jan 2, 2012}\\
\small Mathematics Subject Classifications: 05C10, 05C30, 05C45}

\begin{document}

\maketitle

\begin{abstract}
In this paper the number
of 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, the
number of nonroot-vertex-blocks. As special cases, formulae of several types of rooted $4$-regular
maps such as $2$-connected $4$-regular projective planar maps,
rooted $2$-connected (connected) $4$-regular projective planar maps
without loops are also presented. Several known results on the number
of $4$-regular maps on the projective plane are also concluded. Finally,
by use of Darboux's method, very nice asymptotic formulae for
the numbers of those types of maps are given.
\end{abstract}

\section{Introduction}

We follow [7, 18, 31] to define a graph (map).
A graph (map) is connected and may have loops or multi-edges
(or {\it parallel edges} as some people called it). A graph (map) is
{\it $k$-connected} if it needs at least $k$ vertices to separate
the graph (map) [7]. One may see that this definition is slightly
different from that given by Tutte [30]. For instance, a $2$-connected
graph (map) may have loops which have been excluded by Tutte.

A {\it planar map} ({\it projective planar map}) is a graph $G$
drawn on the sphere $S_0$ (the projective plane $N_1$) such that
edges intersect at vertices and each component
of $S_{0}-G(N_1)$ is a disc called {\it face}. Generally, we may
define a map on higher surfaces. A circuit $\cal C$ on a surface
$\Sigma$ is {\it essential} (or {\it noncontractible} as some people
named it) if $\Sigma-\cal C$ has no component homeomorphic to an
open disc. Otherwise it is called {\it planar} (or {\it trivial}).

A map $M$ is {\it rooted} if an edge $e_r(M)$, a vertex $v_r$ on the edge and a
direction along one side of the edge are all distinguished. All maps
here are rooted unless special statements are given. One may see from
some pioneer's works [3, 18, 31] that rooting a map may trivialize the
{\it automorphism group} and makes it possible to build rooted maps
by recursive relations. So the general way for one to count rooted
maps of a given type in an exact manner is to set up functional equation(s),
usually the {\it order} of the equation(s) goes higher with the increase
of the {\it genera} of the surfaces or some other restrictions (as
the connectivity etc.) on the maps (graphs), and then crack the
equation(s) in every way. The most successful way of doing so is the
so-called {\it quadratic method} (or {\it double-root method} as
some people named it) developed by Brown [8] which enables one to solve
a quadratic equation by solving a system of two or more equations.
This method almost fails in the maps on higher surfaces. In view
of exact enumerating nonplanar maps, some people such as  T.W.S. Walsh et al. [35]
did some works in a general way. Both D. Arqu$\grave{e}$s [1] and, independently,
E.A. Bender et al. [4, 5] counted rooted maps on the torus as a
function of the number of edges. Moreover, Gao et al. [13] treated the exact
enumeration of rooted 3-connected triangular maps on the projective
plane and obtained a simple parametric expression for its generating
function of the number of vertices. Since elegant formulae were very
difficult to obtain for maps on general surfaces, some people
such as E.A. Bender started systematically working on asymptotic formulae.
Many scholars such as E.A. Bender et al. [6], G. Chapuy et al. [10], Gao [11, 12],
A. Mednykh et al. [22, 23] and T.W.S. Walsh et al. [36] have investigated many types
of maps on general surfaces and gotten asymptotic evaluations of nonplanar
maps up to now. For a survey one may see [3] or [17].

A {\it $($rooted$)$ near-$4$-regular map} is such one having
all the vertices $4$-valent except possibly the rooted one. It is
clear that a near-$4$-regular map is {\it Eulerian}. A map is called
{\it near-simple} if no loops or multi-edges are permitted except
possibly only two parallel edges containing the root-vertex.

$4$-regular maps are very important for applications in many fields
such as {\it rectilinear embedding} in VLSI, the {\it Gaussian crossing
problem} in graph theory, the {\it knot problem} in topology and the
enumerations of some other types of maps [18--21]. Rooted (near-)
$4$-regular maps (or their dual: {\it quadrangulations}) have been
investigated by many scholars. We list them (as far as we know) as
follows:

\noindent (1) {\it rooted bicubic maps} [32];\\
\noindent (2) {\it rooted trees} [33];\\
\noindent (3) {\it rooted quadrangulations} [9];\\
\noindent (4) {\it rooted c-nets via quadrangulations} [24];\\
\noindent (5) {\it rooted one-faced maps on surfaces} [35, pp212--213];\\
\noindent (6) {\it rooted $2d-$regular maps on all surfaces} [15];\\
\noindent (7) {\it rooted $4$-regular planar maps} [18, pp159--166];\\
\noindent (8) {\it rooted near-$4$-regular planar Eulerian trials} [27];\\
\noindent (9) {\it rooted loopless $4$-regular maps on the projective plane,
torus and the Klein bottle} [25--28];\\
\noindent (10) {\it rooted $2$-connected $4$-regular maps on the plane} [29].

We expect that several more classes of $4$-regular maps could be added
into this list. This is main aim of this paper.

\noindent{\bf Remark.}\quad Here we regard planar trees or more generally: maps
with one face on surfaces, some people also called them {\it monopoles},
as a special kind of near-$4$-regular maps.

\section{A general equation for maps on $N_1$}

In this section we shall set up a general equation with up to six more
parameters for rooted near-$4$-regular maps on the projective plane
which will imply several new results for some classes of maps unhandled
before and conclude several known results cited in the list above. But
first we should give some more definitions on maps.

Let $\cal U$ and ${\cal U}_p$, respectively, denote the set of all
rooted near-4-regular maps on the plane and the projective plane. Let
their {\it enumerating functions} be, respectively,
\begin{align*}
f(x,y,z,t,w,q)&=\sum_{M\in{\cal U}}x^{2m(M)}y^{s(M)}z^{n(M)}
t^{\alpha(M)}w^{\beta(M)},\\
f_p(x,y,z,t,w,q)&=\sum_{M\in{{\cal U}_p}}x^{2m(M)}y^{s(M)}z^{n(M)}
t^{\alpha(M)}w^{\beta(M)},
\end{align*}
where the variables $x, y, z, t, w$ and $q$ mark, respectively, the
root-valency, the number of edges, the number of inner faces, the number
of nonroot-vertex-loops, the number of {\it cut-vertices} other than the
rooted one.

The set ${\cal U}_p$ may be partitioned into three parts as
\begin{align*}
{\cal U}_p={\cal U}_{p1}+{\cal U}_{p2}+{\cal U}_{p3},
\end{align*}
where
\begin{align*}
{\cal U}_{p1}&=\{M| e_r(M)\ is\ a\ planar \ loop\},\\
{\cal U}_{p2}&=\{M| e_r(M)\ is\ an\ essential\ loop\},\\
{\cal U}_{p3}&=\{M| e_r(M)\ is\ a\ link\}.
\end{align*}

\noindent{\bf Lemma 1.}\quad {\it Let ${\cal U}_{<p1>}=\{M-e_r(M)
|M\in{{\cal U}_{p1}}\}$. Then
\begin{align*}
{\cal U}_{<p1>}={\cal U}\odot{\cal U}_p+{\cal U}_p\odot{\cal U},
\end{align*}
where ``$\odot$'' is the $1v$-production of the sets of
maps defined in} $[18, pp88-89]$.

\noindent{\it Proof.}\quad For a map $M\in{\cal U}_p$, the root-edge $e_r(M)$
is a planar loop. The inner and outer regions determined by $e_r(M)$
are, respectively, two elements of $\cal U$ and ${\cal U}_p$. Since this
procedure is reversible, the lemma follows.$\hfill\Box$

By the above lemma, the enumerating function of ${\cal U}_1$ is
\begin{align}
f_{p1}=2x^2yzff_p.
\end{align}

Since maps in ${\cal U}_{p2}$ are in fact obtained from planar ones
by making a {\it twist} at the root-vertex and then introducing
a loop with only one side and the operation is reversible, after
a similar procedure as [25] we have the enumerating function of
${\cal U}_{p2}$ as
\begin{align}
f_{p2}=x^2y\frac{\partial (xf)}{\partial x}.
\end{align}

The following result is easy to obtain from the definition.

\noindent{\bf Lemma 2.}\quad  {\it Let ${\cal U}_{(p3)}=\{M\bullet e_r(M)
|M\in{{\cal U}_{p3}}\}$. Then ${\cal U}_{(p3)}={\cal U}_p-{\cal U}_p(2)$,
where ${\cal U}_p(2)$ is the set of maps in ${\cal U}_p$ whose
root-valencies are all $2$, where $M\bullet e_r(M)$ denotes the rooted map after contracting the root-edge
$e_r(M)$ of a map $M$}.

By Lemma 2, the enumerating function of ${\cal U}_{(p3)}$ is $f_{(p3)}=f_p-x^2F_p(2)$,
where $F_p(2)$ is the enumerating function of ${\cal U}_p(2)$.

Since splitting the root-vertex may create nonroot-vertex loops,
the set ${\cal U}_{(p3)}$ has to be divided into several more parts as
\begin{align*}
{\cal U}_{(p3)}=\sum_{i=1}^6{\cal U}^i_{(p3)},
\end{align*}
where maps in ${\cal U}^i_{(p3)} (1\leq i\leq 5) $ have the structures
as depicted in Fig. 1, where the shadowed regions represent a map on $N_1$ or $S_0$.
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\put(65,35){$M\in{{\cal U}_{(p3)}^4}$}
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\put(42,25){Fig. 1\quad(Five types of maps which will induce a nonroot-vertex}
\put(57,20){ loop after splitting the root-vertex.)}
\end{picture}
\end{center}

\noindent{\bf Remark.}\quad (1) Maps of type 1 (or 2) have their edge
$e_r(M)$ (or $e_{Pr}(M)$) the planar loop, here $P$ is the rotation
of $M$ at the root-vertex; (2) Maps of the above types will not
create nonroot-vertex after splitting the root-vertex.

There are 5 cases corresponding to the dividing of ${\cal U}_{(p3)}$
which must be handled.

\noindent{\bf Case 1.}\quad {\bf The maps of ${\cal U}_{(p3)}^1$.}

Notice that for a map of ${\cal U}_{(p3)}^1$, two things will definitely
happen: the number of nonroot-vertex loops will increase and the number
of nonroot-cut-vertices will not change after splitting the root-vertex.
It is easy to see that the contributions of ${\cal U}_{p3}^1$ to ${\cal U}_p$
is
\begin{align}
f_{p3}^1=ty^2zf_p.
\end{align}

\noindent{\bf Case 2.}\quad{\bf The maps of ${\cal U}_{(p3)}^2$.}

Under this condition, the edge $e_{Pr}(M)$ is always a planar loop
while the root-edge $e_r(M)$ is a loop which will create a
nonroot-vertex loop. This situation is very similar to the previous case except
that $e_r(M)$ may be an essential loop. Thus, we find that the enumerating
function of ${\cal U}_{p3}^2$ is exactly equal to $f_{p3}^1$, i.e.,
\begin{align}
f_{p3}^2=ty^2f_p.
\end{align}

\noindent{\bf Case 3.}\quad{\bf The maps of ${\cal U}_{(p3)}^3$.}

Similar to what we have done in Cases 1 and 2, one may partition the
set ${\cal U}_{(p3)}^3$ into two parts as
\begin{align*}
{\cal U}_{(p3)}^3={\cal U}_{(p3)}^{31}+{\cal U}_{(p3)}^{32},
\end{align*}
where ${\cal U}_{(p3)}^{31}=\{M | \rm both\,\,e_r(M)\,\,and\,\,e_{P^{2}r}(M)\
are\ loops\}$.

Hence maps in ${\cal U}_{(p3)}^{31}$ have the following structure in
the left side of Fig. 2.
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Thus, the enumerating function of ${\cal U}_{p3}^3$ is
\begin{align}
f_{p3}^3=ty^2(f-1).
\end{align}

\noindent{\bf Case 4.}\quad{\bf The maps of ${\cal U}_{(p3)}^4$.}

As we have reasoned previously, we partition the set ${\cal U}_{(p3)}^4$
into two parts as
$$
{\cal U}_{(p3)}^4={\cal U}_{(p3)}^{41}+{\cal U}_{(p3)}^{42},
$$
where ${\cal U}_{(p3)}^{41}=\{M | \rm both\,\,e_r(M)\,\,and\,\,e_{Pr}(M)\,\,
are\,\,crossing\,\,essential\,\,loops\}$
and maps of ${\cal U}_{(p3)}^{41}$ have a configuration as shown in
the center of Fig. 2. Therefore the contribution of ${\cal U}_{(p3)}^{41}$ is $f_{(p3)}^{41}=x^4y^2zf^2$
which implies
\begin{align}
f_{p3}^{4}=ty^2(f-1).
\end{align}

\noindent{\bf Case 5.}\quad{\bf The maps of ${\cal U}_{(p3)}^5$.}

Also the set ${\cal U}_{(p3)}^5$ should be divided in a way of
$$
{\cal U}_{(p3)}^5={\cal U}_{(p3)}^{51}+{\cal U}_{(p3)}^{52},
$$
where ${\cal U}_{(p3)}^{51}=\{M | e_r(M)\rm\,\,and\,\,e_{Pr}(M)\,\,
are\,\,two\,\,types\,\,of\,\,loops\}$.

It is clear that the two edges $e_r(M)$ and $e_{Pr}(M)$ are in a
position as in the right side of Fig. 2. Now one may readily see that the enumerating function of
${\cal U}_{p3}^5$ is
\begin{align}
f_{p3}^5=ty^2(f-1).
\end{align}

\noindent{\bf Case 6.}\quad{\bf The maps of ${\cal U}_{(p3)}^6$.}

Before investigating the contribution of ${\cal U}_{(p3)}^6$, we have to
state some basic facts.

\noindent{\bf Fact 1.}\quad {\it For a map $M$ of ${\cal U}_{(p3)}^6$,
splitting the root-vertex will not create nonroot-vertex loops but may
increase the number of nonroot-cut-vertices.}

Consequently, we have to divide the set ${\cal U}_{(p3)}^6$ into
two parts as
$$
{\cal U}_{(p3)}^6={\cal U}_{(p3)}^{61}+{\cal U}_{(p3)}^{62},
$$
where $M\in{{\cal U}_{(p3)}^{61}}\Longleftrightarrow$ splitting the
root-vertex will create a nonroot-cut-vertex.

Next we study the structures of the maps in ${\cal U}_{(p3)}^{61}$.
Since the splitting-procedure defined previously will create a
nonroot-cut-vertex, say $v_r^{''}$, we conclude that $v_r^{''}$ will
connect a map other than the {\it vertex-map} (i.e., the map consists
of a single vertex) whose root-valency is $2$. Since there are two
types of such $2$-valent maps attached to the vertex $v_r^{''}$, the set
${\cal U}_{(p3)}^{61}$ should be partitioned into three parts as
$$
{\cal U}_{(p3)}^{61}={\cal U}_{(A)}+{\cal U}_{(B)}+{\cal U}_{(C)},
$$
where maps of those types are depicted in Fig. 3.
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\put(28,70){Fig. 3\quad(Three types of maps in ${\cal U}_{(p3)}^{61}$
which will create a}
\put(43,64){ nonroot-cut-vertex after splitting the root-vertex)}
\end{picture}

\end{center}

\noindent{\bf Lemma 3.}\quad {\it The set ${\cal U}_{(A)}$ is a composition of
two types of maps, one is planar while the other a nonplanar map, i.e.,
$$
{\cal U}_{(A)}=({\cal U}_p(2)-L_p)\odot({\cal U}-\vartheta)+
({\cal U}(2)-L)\odot{\cal U}_{p},
$$
where ${\cal U}_p(2)$ and ${\cal U}(2)$ are, respectively, the set of
maps of ${\cal U}_p$ and $\cal U$ whose root-valencies are all $2$.
Meanwhile, $L_p$ and $L$ are, respectively, the loop map on the
projective plane and the sphere}.

Furthermore, by considering the embedding ways of the 2-valent maps
attached to the nonroot-cut-vertices, the maps of ${\cal U}_{(A)}$
are classified into two types of maps as defined  in Fig. 4,
where the shadowed regions are either in
${\cal U}_p(2)\cup{\cal U}(2)$ or ${\cal U}\cup{\cal U}_p$.
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Hence, the enumerating function of ${\cal U}_{(A)}$ is
\begin{align}
f_{(A)}=x^2(F_p(2)-y)(f-1)+x^2(F(2)-yz)f_p,
\end{align}
where $F_p(2)$ and $F(2)$ are, respectively, the enumerating function
of ${\cal U}_p(2)$ and ${\cal U}(2)$.

It is clear that the three shadowed regions in the maps in the left
side of Fig. 5 are, respectively, two planar maps and a nonplanar one.
Thus, the contribution of such types of maps is $x^{4}yz(F_p(2)-y)f^{2}$
and hence the other types of maps of $({\cal U}_p(2)-L_p)
\odot(\cal U-\vartheta)$ will contribute
$$
x^2(F_p(2)-y)(f-1)-x^4yz(F_p(2)-y)f^2.
$$
So we have

\noindent{\bf Lemma 4.}\quad {\it The contribution of $({\cal U}_p(2)-L_p)
\odot(\cal U-\vartheta)$ to ${\cal U}_p$ is $wy(F_p(2)-y)(f-1)$}.

We now turn to the second types of maps in Lemma 3. Unlike its
counterpart, the set $({\cal U}(2)-L)\odot{\cal U}_p$ consists
of three types of maps among which two are defined, respectively,
in the center and right parts of Fig. 5.

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\end{center}

It is easy to see that those two types of maps stated above will,
respectively, contribute
$$
x^4y(F(2)-yz)\frac{\partial (xf)}{\partial x}\quad{\rm and}\quad x^4yz(F(2)-yz)f_p
$$
to ${\cal U}_{(B)}$. Thus, we may obtain the following

\noindent{\bf Lemma 5.}\quad {\it The contribution of $({\cal U}(2)-L)
\odot{\cal U}_p$ to ${\cal U}_p$ is $\frac{yw}{x^2}(x^2(F(2)-yz)f_p)$}.

Now we begin to handle the set ${\cal U}_{(B)}$. Similar to Lemma 3,
we may have

\noindent{\bf Lemma 6.}\quad {\it The set ${\cal U}_{(B)}$ is the composition
of two types of maps, one is a planar map while the other is nonplanar,
i.e.,
$$
{\cal U}_{(B)}=({\cal U}_p(2)-L_p)\otimes({\cal U}-\vartheta)+
({\cal U}(2)-L)\otimes{\cal U}_p,
$$
where the operation "$\otimes$" of maps is defined in} Fig. 6.
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\end{center}


After an analogous procedure in our proofs of Lemmas 4 and 6, we
obtain the following

\noindent{\bf Lemma 7.}\quad {\it The contribution of $({\cal U}_p(2)-L_p)
\otimes({\cal U}-\vartheta)$ to ${\cal U}_p$ is $\frac{yw}{x^2}(x^2(F_p(2)-y)(f-1))$}.

\noindent{\bf Lemma 8.}\quad {\it The contribution of $({\cal U}(2)-L)
\otimes{\cal U}_p$ to ${\cal U}_p$ is $\frac{yw}{x^2}(x^2(F(2)-yz)f_p)$}.


Similar to what we have reasoned in Lemmas 4, 5, 7 and 8, the set
${\cal U}_{(C)}$ should be partitioned into two types of maps according
to whether the root-vertex splitting-procedure will create a pair
of multi-edges or not. The first kind of maps has a configuration
in Fig. 7, where the shadowed region represented by a perpendicular
essential circuit is a map of ${\cal U}(2)-L$ embedded in $N_1$, the
projective plane, while the other two indicate planar maps.
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\noindent{\bf Lemma 9.}\quad {\it The contribution of ${\cal U}_{(C)}$ to
${\cal U}_p$ is $\frac{yw}{x^2}\left\{\frac{x^2}{z}(F(2)-yz)(f-1)\right\}$}.

Now, we begin to establish the main result of this section. By (1) and (2), we have
\begin{align}
f_p=2x^2yzff_p+x^2y\frac{\partial(xf)}{\partial x}+f_{p3},
\end{align}
where
\begin{align}
f_{p3}&=\frac{y}{x^2}\{f_p-x^2F_p(2)\}
+\sum_{i=1}^5\left(f_{p3}^i-\frac{y}{x^2}f_{(p3)}^i\right)\nonumber\\
&\quad+\left(f_{p3}^{61}-\frac{y}{x^2}f_{(p3)}^{61}\right)
+\left(f_{p3}^{62}-\frac{y}{x^2}f_{(p3)}^{62}\right).
\end{align}

By equations (3)-(8),
\begin{align*}
f_{p3}^1-\frac{y}{x^2}f_{(p3)}^1&=y^2z(t-1)f_p,\\
f_{p3}^2-\frac{y}{x^2}f_{(p3)}^2&=y^2z(t-1)f_p,\\
f_{p3}^3-\frac{y}{x^2}f_{(p3)}^3&=(t-1)y^2(f-1),\\
f_{p3}^4-\frac{y}{x^2}f_{(p3)}^4&=(t-1)y^2(f-1),\\
f_{p3}^5-\frac{y}{x^2}f_{(p3)}^5&=(t-1)y^2(f-1),\\
f_{p3}^{61}-\frac{y}{x^2}f_{(p3)}^{61}&=
\left\{f_A-\frac{y}{x^2}f_{(A)}\right\}+
\left\{f_B-\frac{y}{x^2}f_{(B)}\right\}+
\left\{f_C-\frac{y}{x^2}f_{(C)}\right\},\\
f_A-\frac{y}{x^2}f_{(A)}&=(w-1)y(F_p(2)-y)(f-1)+(w-1)y(F(2)-yz)f_p,\\
f_B-\frac{y}{x^2}f_{(B)}&=(w-1)y(F_p(2)-y)(f-1)+(w-1)y(F(2)-yz)f_p,\\
f_C-\frac{y}{x^2}f_{(C)}&=\frac{(w-1)y}{z}(F(2)-yz)(f-1).
\end{align*}

By the principle of Inclusion-Exclusion, the contribution of
$f_{(p3)}^{62}$ is
$$
f_{(p3)}^{62}=\sum_{i=1}^3\mid R_i\mid-\sum_{1\leq i<j\leq 3}
\mid R_i\cap R_j\mid+\mid R_1\cap R_2\cap R_3\mid,
$$
where $\mid A\mid$ is used to denote the contribution of $A$ and
\begin{align*}
R_1&=\{M | e_r(M)\  is\ a\ loop\},\\
R_2&=\{M | e_{Pr}(M)\  is\ a\ loop\},\\
R_3&=\{M | e_{P^{2}r}(M)\ is\ a\ loop\}.
\end{align*}

One may foresee from the possible embedding ways of loops on $N_1$
that the calculations of $f_{(p3)}^{62}$ is very complicated. But
its value is irrelevant for our further calculations since
$$
f_{p3}^{62}=\frac{y}{x^2}f_{(p3)}^{62}.
$$

By substituting (10) and the above expressions into (9), we get our first
main result:

\noindent{\bf Theorem A.}\quad {\it The enumerating function of rooted
(2-connected) near-4-regular maps on the projective plane satisfies
the following equation}:
\begin{align*}
f_p&=2x^2yzff_p+x^2y\frac{\partial(xf)}{\partial x}+2y^2z(t-1)f_p+3(t-1)y^2(f-1)\\
&\quad+2(w-1)y(F_p(2)-y)(f-1)+2(w-1)y(F(2)-yz)f_p\\
&\quad+\frac{(w-1)y}{z}(F(2)-yz)(f-1).
\end{align*}

\noindent{\bf Remark.}\quad We have shown in [29] that the function $f$ defined in
Theorem A is also algebraic, i.e., satisfying a polynomial equation, i.e.,
\begin{align}
af^2+bf+c=0,
\end{align}
in which the coefficients of $f^0, f$ and $f^2$ can be expressed as
\begin{align*}
a&=x^4yz,\\
b&=y-x^2+2(t-1)y^2zx^2+2x^2y(w-1)(F_2-yz),\\
c&=x^2-y-x^2yF_2-2(t-1)x^2y^2z-2x^2y(w-1)(F_2-yz).
\end{align*}

If we rearrange the items of the equation in Theorem A, then the
function $f_p$ can be expressed as the functions of $f, F(2)$ and
$F_p(2)$, i.e.,
\begin{align*}
-\sqrt{\triangle}f_p&=x^4y\frac{\partial(xf)}{\partial x}+3(t-1)x^2y^2(f-1)+2(w-1)x^2y(F_p(2)-y)(f-1)\\
&\quad+\frac{(w-1)yx^2}{z}(F(2)-yz)(f-1)-x^2yF_p(2),
\end{align*}
where $\triangle$ is the discriminant of (11), i.e.,
\begin{align*}
-\sqrt{\triangle}&=x^2-2x^4yzf-y-4x^4y^3zt(q-1)f-2yt(t-1)x^2yz\\
&\quad-2(w-1)x^2y(F(2)-yz).
\end{align*}

\section{Calculations}

In this section we shall deal with various types of rooted
(near-) $4$-regular on the projective plane. Now the equation
in Theorem A becomes
\begin{align*}
-\sqrt{\triangle}{f_p}&=x^4y\frac{\partial(xf)}{\partial x}+3(t-1)x^2y^2(f-1)
+2(w-1)x^2y(F_p(2)-y)(f-1)\\
&\quad+\frac{(w-1)yx^2}{z}(F(2)-yz)(f-1)
-x^2yF_p(2),
\end{align*}
where
$$
-\sqrt{\triangle}=x^2-2x^4yzf-y-2(t-1)x^2y^2z-2(w-1)x^2yz(F(2)-yz).
$$

Now, we are in a position of implying the famous {\it double-root method}
developed by Brown [8] since the most important part of $f_p$, the
discriminant of (11), has become a quadratic polynomial of $x^2$.
Let $x^2=\eta$ denote a double root of $\triangle$. Then we may
obtain a system of equations:
\begin{align}
&{\eta}-2{\eta}^2yzf-y-2(t-1)\eta y^2z-2(w-1)\eta y(F(2)-yz)=0,\nonumber\\
&x^2\frac{\partial(xf)}{\partial x}+3(t-1)y(f-1)+
2(w-1)(F_p(2)-y)(f-1)\nonumber\\
&+\frac{(w-1)}{z}(F(2)-yz)(f-1)-F_p(2)=0.
\end{align}

Let $t=w=1$. Then we get general rooted maps on $N_1$ and a very simple
expression for $F_p(2)$:
$$
F_p(2)=\left\{x^2\frac{\partial(xf)}{\partial x}\right\}
{\left|\frac{}{}\right.}_{x^2=\eta},
$$
which will imply the following known result:

\noindent{\bf Corollary 1} ([25]){\bf.}\quad {\it The number of rooted $4$-regular maps
(which may have loops) on the projective plane with $2p-2$ edges is
\begin{align*}
&\sum_{{{m\geq 2}\atop{n\geq 0}}\atop{m+n\leq p}}
\frac{2^{n+1}3^{p-m-n}(m+n)}{mp}\binom{2m-2}{m-1}\binom{m+n-2}{m-2}\\
&\times\binom{2p-m-n-1}{p-1}+\frac{3^{p-1}}{p}\binom{2p-2}{p-1}.\\
\end{align*}}
\noindent{\bf Corollary 2} ([14, 26]){\bf .}\quad {\it The enumerating function of rooted
one-faced maps on the projective plane is}
$$
\frac{1}{\sqrt{1-4x^2}}\sum_{m\geq 1}\binom{2m-1}{m}x^{2m}.
$$

If we consider the case of loopless maps (i.e., $t=0$), then we have
the following

\noindent{\bf Corollary 3} ([26]){\bf.}\quad {\it The enumerating function of rooted
loopless $4$-regular maps on the projective plane is
\begin{align*}
&\sum_{m\geq 2}(A_m+\frac{3}{2}Y_{m-1}-3X_{m-1}-X_m)(1-2y^2)y^{2m}\\
&-\sum_{m\geq 2}3B_{m-1}y^{2m},
\end{align*}
in which}
\begin{align*}
A_m&=\sum_{{l,p\geq 0}\atop{k\geq 2}}\sum_{{s+p=m+1}\atop{s\geq k+2}}
\frac{(-1)^p2^{l+p+1}3^{s-k-l}(k+l)}{k\times s}\binom{2k-2}{k-1}\\
&\quad\times\binom{k+l-2}{l}\binom{2s-k-l-1}{s-1}\binom{2s+p-2}{p},\\
B_m&=\sum_{l+n=m}\frac{(-1)^l2^l3^n}{(n+1)(2n+1)}\binom{2n+2}{n}
\binom{n+m}{l},\\
X_m&=\sum_{l+n=m}\frac{(-1)^l2^{l+1}3^n}{n+1}\binom{2n}{n}
\binom{n+m}{l},\\
Y_m&=\sum_{l+n=m}\frac{(-1)^l2^{l+3}3^n}{n+2}\binom{2n+1}{n}
\binom{n+m}{l}.
\end{align*}

Next, we start to investigate the number of $2$-connected $4$-regular
maps on the projective plane. We first consider general $2$-connected
maps (which may have loops).

\noindent{\bf Case 1.}\quad{\bf General 2-connected 4-regular maps on $N_1$}.

Let $t=y=1$ and $w=0$. Then all the cut-vertices disappear except
for those on the root-vertices by our definition. Furthermore, let
$x^2=\eta$ be a double root of $\triangle$, the discriminant of (11).
Then all the functions and variables in this
case are handled under those conditions unless special statements are
given. Now the function $F_p(2)$ can be expressed in terms of $f$ and
$F(2)$, i.e.,
$$
(2f-1)F_p(2)=\left\{x^2\frac{\partial(xf)}{\partial x}
-\frac{F}{z}(f-1)+2(f-1)\right\}{\left|\frac{}{}\right.}_{x^2=\eta},
$$
where $F=F(2)-z$.

In order to determine the function $F_p(2)$ completely, we have to
state some facts as our step-stones.

\noindent{\bf Fact 2} ([29]){\bf .}\quad {\it The function $F$ may be expressed in terms
of $\eta$ and $z$, i.e.,
$$
F=\frac{1-\eta+2{\eta}^3z(1-z)}{2\eta(1-{\eta}^2z)},
$$
in which}
\begin{align}
\eta=1+\frac{z}{2}\phi(\eta,z),\quad
\phi(\eta,z)=\frac{{\eta}^2(1+\eta-2{\eta}z)^2}
{(1+2\eta z)(1-{\eta}^2z)}.
\end{align}

Since the function $f$ is algebraic, after some manipulations of the
system of equations (12) we may find the following

\noindent{\bf Fact 3.}\quad {\it The functions $F$ and $f$ are related in such a
way that
$$
\eta-2{\eta}^2zf-1+2\eta F=0,\quad
\eta\frac{\partial(xf)}{\partial x}=\frac{1-\sqrt{1-v\eta}}{\eta z}
-\eta f,
$$
where} $f=\frac{1}{1-{\eta}^2zf}$.

 The Lagrangian inversion formula stated above is vital in our simplification of the parametrical expression
of $F_p(2)$. In fact, another inversion formula cited in Fact 2 may
also be expressed in terms of $f$, i.e.,
\begin{align}
\eta=1+\frac{z}{2}\phi(\eta,z),\quad
\phi(\eta,z)=\eta f\left\{1+\frac{\eta f}{1-f}\right\}.
\end{align}

Based on the Facts 2 and 3, the function $F_p(2)$ can be expressed
by $\eta$ and $f$ elegantly, i.e.,

\noindent{\bf Fact 4.}\quad {\it The enumerating function of rooted
$2$-valent near-$4$-regular maps (which may have loops) without
cut-vertices on the projective plane is
$$
(2f-1)F_p(2)=\frac{1-\sqrt{1-v\eta}}{\eta z}
+\frac{\eta(\eta-3)}{2}f^2+2(f-1),
$$
in which $v\eta=4{\eta}^2zf$.}

Although $F_p(2)$ presents a formula to count a type of rooted
$2$-connected $4$-regular maps on $N_1$, we may readily see from the
definition that if loops are permitted, then the enumerating function for
rooted $2$-connected $4$-regular maps on $N_1$ depends on both
$F_p(2)$ and the same type of maps on the plane. In fact, rooted
$2$-connected $4$-regular maps on $N_1$ consist of $5$ types of maps
which are depicted in Fig. 8.

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\put(40,86){Fig. 8\quad (Five types of $2$-valent maps which will result in}
\put(55,81){  five kinds of $4$-regular maps on $N_1$)}
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\end{center}

\noindent{\bf Fact 5.}\quad {\it The enumerating function of rooted $2$-connected
$4$-regular maps on $N_1$ can be expressed as
$$
F_p(4)=(1+2z)(F_p(2)-1)+3F,
$$
where $F_p(2)$ and $F$ are defined in Facts $2$ and $4$}.

By using Lagrangian inversion (for a reference, one may see [16])
for these in (13) and Fact 3, the two functions may be expanded into a power
series of the variable $z$, i.e.,

\noindent{\bf Theorem B.}\quad {\it The enumerating function of rooted $4$-regular
maps (which may have loops) without cut-vertices on the projective plane
is
\begin{align*}
F_p(4)&=\sum_{{k\geq 0}\atop{n\geq k}}\frac{2^kk}{n}
\binom{2n-k-1}{n-1}(1+2z)z^{n-k}{\eta}^{2(n-k)}\\
&\quad+\sum_{{k\geq 0}\atop{n\geq k+2}}\frac{2^{k-1}(k+2)}{n}
\binom{2n-k-3}{n-1}(1+2z)z^{n-k-2}\\
&\quad \times (\eta-3){\eta}^{2(n-k-2)+1}\\
&\quad+\sum_{{k\geq 1}\atop{l\geq 0}}\sum_{n\geq k+l}
\frac{2^{l+1}(k+l)}{kn}\binom{2k-2}{k-1}\binom{2n-k-l-1}{n-1}\\
&\quad \times (1+2z)z^{n-l-1}{\eta}^{2(n-l)-1}\\
&\quad+3\sum_{{m\geq 2}\atop{n\geq m}}\frac{m}{2n}\binom{2n-m-1}{n-1}
z^{n-m+1}(\eta-1){\eta}^{2(n-m)+1}\\
&\quad +3\sum_{n\geq 1}\frac{1}{n}\binom{2n-2}{n-1}z^n{\eta}^{2n-1},
\end{align*}
where}
\begin{align*}
{\eta}^s&=\sum_{m\geq 1}\sum_{p=0}^m\sum_{{q\geq 0}\atop{n\geq m+p+q}}
\frac{s(m+p+q)}{m!n}\binom{2n-m-p-q-1}{n-1}\\
&\quad\times\binom{p+q-1}{p-1}D^{(m-1)}_{\eta=1}
\left\{{\eta}^{2n-m-p-2q}\right\}z^{n-q},
\end{align*}
{\it in which the differential operator ``$D^{(m-1)}_{\eta=1}$'' is defined as}
$$
D^{(m-1)}_{x=a}H=\frac{{\partial}^{m-1} H}{\partial x}
{\left|\frac{}{}\right.}_{x=a}.
$$

By using an algebraic symbolic system, say {\bf\it MAPLE}, the first fewer
coefficients of the function $F_p(4)$ may be calculated:
\begin{align*}
F_p(4)&=5z+38z^2+199z^3+1466z^4+12365z^5+109700z^6\\
&\quad +1003929z^7+9404402z^8+\cdots
\end{align*}

One may see that the low values of maps coincide with our calculation.
For instance, there are $38$ distinct rooted $2$-connected $4$-regular
maps with $4$ edges on the projective plane which are determined by the
maps shown in Fig. 9.
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\put(55,38){Fig. 9\quad(Six unrooted maps on $N_1$ which will}
\put(72,33){determine $38$ distinct rooted maps)}

\end{picture}

\end{center}


The contributions of the $6$ maps to the rooted maps are, respectively,
listed in the following table:
$$
\begin{tabular}{|c|c|c|c|c|c|}\hline
$M_1$ & $M_2$ & $M_3$ & $M_4$ & $M_5$ & $M_6$ \\ \hline
8 & 8 & 8 & 2 & 4 & 8 \\ \hline
\end{tabular}
$$

\noindent{\bf Remark.}\quad The results stated previously permit the
existence of loops and maps there are slightly different from what
defined by Tutte [30] where a $2$-connected graph (map) contains no
loops. This is just the situation we shall handle next.

\noindent{\bf Case II.}\quad{\bf $2$-connected $4$-regular maps without loops.}

Let $y=1$ and $t=w=0$. Then all the loops and cut-vertices are
eliminated except for those on the root-vertices. Still let $x^2=\eta$
be a double root of the discriminant of (11). Then all the functions and
variables appearing in this case are all under those restrictions unless
special statements are given. Similar to what we have stated previously,
we have

\noindent{\bf Fact 6} ([29]){\bf .}\quad {\it The function $F$ may be expressed as a function
of $\eta$ and $z$, i.e.,
\begin{align}
F&=\frac{1-\eta-2\eta z+2{\eta}^3z(1+z)}{2\eta(1-{\eta}^2z)},\nonumber\\
\eta&=1+z\phi(\eta,z),\quad \phi(\eta,z)={\eta}^2\left\{\frac{(1+\eta)^2}
{2(1-{\eta}^2z)}-2\right\},
\end{align}
in which $F=F(2)-z$}.

\noindent{\bf Fact 7.}\quad {\it The function $F$ and $f$ are related in such a
way that
$$
\eta-2{\eta}^2zf-1+2\eta z+2\eta F=0,\quad
\eta\frac{\partial(xf)}{\partial x}=\frac{1-\sqrt{1-v\eta}}{\eta z}
-\eta f,
$$
where}
\begin{align}
f=\frac{1}{1-{\eta}^2zf}.
\end{align}

\noindent{\bf Remark.}\quad There are several things striking us most:
(1) The formula (16) is almost the same as the one for
trees presented by Tutte [33] and this raises a natural question: Can we
construct planar (near-)$4$-regular maps from trees ? (2)
Eliminating loops will not change some properties cited in the
Facts 2 and 3, especially the formula in Fact 3 (which is the same as (16)).
But from the results in [29] one may see that those two types of
$2$-connected $4$-regular maps on the plane are indeed different in
essentiality since we have shown there that their convergence
radius are distinct. Instead, as our discussion goes on, one may
see that such strict condition on maps (i.e., destroying loops) will
make our calculations much more easier and simpler. Besides, the
function $\phi(\eta,z)$ appearing in Fact 5 can be simply expressed
as a function of $f$, i.e.,
\begin{align*}
\phi(\eta,z)&={\eta}^2\left\{\frac{(1+\eta)^2}{2(1-{\eta}^2z)}-2\right\}\\
&=2{\eta}^2f(f-1),
\end{align*}
which will lead to a simpler formula of $F_p(2)$.

Since the procedure is very much alike as we get in Theorem B, we
omit the proofs and only state them here.

\noindent{\bf Theorem C.}\quad {\it The enumerating function of $2$-connected
loopless rooted $4$-regular maps on the projective plane is
$$
F_p(4)=\frac{1-\sqrt{1-v\eta}}{\eta z(2f-1)}
-\frac{\eta f}{2f-1}-\frac{{\eta}^3zf^3(2-f)}{2f-1}-1,
$$
in which $v\eta=4{\eta}^2zf$.}

After applying Lagrangian inversion for the formulae in Facts 4 and 5,
we may expand $F_p(4)$ into a power series of $z$, i.e.,
\begin{align*}
F_p(4)&=\sum_{{k\geq 0}\atop{n\geq k+3}}\frac{2^k3(k+3)}{n}
\binom{2n-k-4}{n-1}z^{n-k-2}{\eta}^{2(n-k-2)+1}\\
&\quad-\sum_{{k\geq 0}\atop{n\geq k+3}}\frac{2^k(k+3)}{n}
\binom{2n-k-4}{n-1}z^{n-k-3}{\eta}^{2(n-k-3)+1}\\
&\quad-\sum_{{k\geq 2}\atop{l\geq 0}}\sum_{n\geq k+l}
\frac{2^{l+1}(k+l)}{kn}\binom{2k-2}{k-1}\binom{2n-k-l-1}{n-1}
z^{n-l-1}{\eta}^{2(n-l)-1}\\
&\quad -1,
\end{align*}
where for $s\geq 1$,
\begin{align*}
{\eta}^s&=1+\sum_{{m\geq 1}\atop{p\geq 0}}\sum_{q\geq 2m+p}
\frac{s(2m+p)}{q(2q+s)}\binom{m+p-1}{p}\\
&\quad \times \binom{2q-2m-p-1}{q-1}\binom{2q+s}{m}z^q.
\end{align*}
After some manipulations via an algebraic system such as {\bf\it MAPLE},
the first fewer coefficients of the function $F_p(4)$ may be determined:
\begin{align*}
F_p(4)&=6z^2+21z^3+138z^4+781z^5+4836z^6\\
&\quad +30099z^7+191698z^8+\cdots
\end{align*}

One may compare the initial values of maps with the number presented
by this formula and see that they are coincide with each other. For
instance, there are $6$ rooted loopless $4$-regular maps with $4$ edges
on $N_1$ as determined by the two maps $M_4$ and $M_5$ in Fig. 10.
Furthermore, the following Fig. 10 shows another group of unrooted maps
which will induce $21$ distinct rooted maps with $4$ faces on $N_1$.
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\put(20,51){Fig. 10\quad(Four distinct embedding of the double graph}
\put(38,46){$K_3^2$ on $N_1$ which will induce $21$ rooted maps)}
\end{picture}
\end{center}

The contributions of the four maps to the rooted maps are, respectively,
listed in the following table:
$$
\begin{tabular}{|c|c|c|c|}\hline
$M_1$ & $M_2$ & $M_3$ & $M_4$  \\ \hline
12 & 6 & 2 & 1  \\ \hline
\end{tabular}
$$

\section{Asymptotic evaluations}

In this section we concentrate on the approximate values of the number
of the two types of maps obtained in the last section. Since applying
of the Darboux's theorem [2, Theorem 4] needs some theory backgrounds,
we have to state some basic facts before our evaluations.

\noindent{\bf Fact 8.}\quad {\it Let $M$ be a class of infinite rooted
maps with $M_1$ as its subset. Suppose that their enumerating functions
may be expanded into power series and their respective convergence
radiuses are $R$ and $R_1$. Then $R$ and $R_1$ are, respectively, the
singularities of them. Furthermore, $R\leq R_1$}.

In the case of $y=t=w=1$, the function $F_p(2)$ has a very simple
expression:
$$
zF_p(2)=1-\frac{3\theta}{2}-(1-\theta)\sqrt{1-\frac{2\theta}{1-\theta}},\quad
\theta=\frac{4z}{2-3\theta}.
$$
By a result in [2, Theorem 5] and Darboux's Theorem [2, Theorem 4], we have the following

\noindent{\bf Fact 9[15].}\quad {\it The convergence radius of the power series
expansion with the number of inner faces as the parameter of the
enumerating function for the rooted $4$-regular maps on the projective
plane is $\frac{1}{12}$ and consequently, the number of rooted 4-regular
maps on the projective plane with $n$ inner faces is asymptotic to
$$
-\frac{C12^n}{n^{\frac{5}{4}}\Gamma(-\frac{1}{4})},
$$
where $C=\frac{\sqrt{6}}{3}$. Furthermore, any type of infinite rooted
$4$-regular maps on the projective plane must have its convergence
radius, say $r$, of enumerating function satisfying}
$$
\frac{1}{12}\leq r \leq 1.
$$

We first consider the asymptotic evaluation of the number of rooted
$2$-connected $4$-regular maps (which may have loops) on $N_1$, i.e.,
$y=t=1$ and $w=0$. So all the discussions in this case are under
those restrictions unless special statements are given. The expressions
in Fact 2 may be rewritten as
\begin{align}
F=\frac{1-x+2x^3z(1-z)}{2x(1-x^2z)},\quad
x=1+\frac{zx^2(1+x-2xz)^2}{2(1+2xz)(1-x^2z)}.
\end{align}

\noindent{\bf Remark.}\quad Here we use $x^2=x=\eta$ to denote the double root
of (11) in convenience.

Since Darboux's Theorem needs the location of the singularity
corresponding to the convergence radius of the enumerating function,
we have to determine the convergence radius of $F_p(4)$ defined in Fact 5.
By definition of $F_p(4)$, one may see the following

\noindent{\bf Fact 10.}\quad  {\it The singularities of $F_p(4)$ satisfy either\\
\noindent $(1)$\quad $2f-1=0$,\\
\noindent $(2)$ or\quad $x=0$,\\
\noindent $(3)$ or\quad $vx=1$,\\
\noindent $(4)$ or\quad Equations in $(20)$}.

It is clear that (17) implies $x\not=0$. If $2f=1$, then by Fact 3 we
have $x=1$ or $1+2xz=0$ either will contradict (17). So the wanted
singularities will be determined by $xv=1$ or equations of (17). The
condition $vx=4x^2zf$ together with (18) shows that the singularities of
$F_p(4)$ will be induced by the following system of equations:
\begin{align}
&4(x-1)(1+2xz)=1+x-2xz,\nonumber\\
&2x^2z(x+1-2xz)=1-x^2z.
\end{align}

The variable $z$ my be extracted from the above equations such that
$$
z=\frac{5-3x}{2x(4x-3)}.
$$
We now substitute it into an equation of (18) and arrive at another
equation of $x$:
\begin{align}
x(5-3x)(4x^2+6x-\frac{9}{2})-(4x-3)^2=0.
\end{align}
By using {\bf\it MAPLE} we may solve the equation by factoring the left
side of (19) as
\begin{align}
(x-1)(x+2)(24x^2-28x-9)=0
\end{align}
and find its zeros and the corresponding values of $z$ as listed
in the following table:
$$
\begin{tabular}{|c|c|c|c|c|} \hline
$x$ & 1 & -2 & $\frac{7+\sqrt{103}}{12}$ & $\frac{7-\sqrt{103}}{12}$  \\ \hline
$z$ & 1 & $\frac{1}{4}$ & 0.0918035245 & 2.723004459  \\ \hline
\end{tabular}
$$

Now, what left for us is to determine the wanted singularity
corresponding to the convergence radius. The following fact is useful
in our evaluations.

\noindent{\bf Fact 11.}\quad {\it The convergence radius of the power series
with the number of inner faces as the parameter for the rooted
$2$-connected $4$-regular planar maps is $0.0918035245$. Furthermore, the
convergence radius, say $r$, of the power series expansion of $F_p(4)$
must satisfy the following condition:
$$
\frac{1}{12}\leq r\leq 0.0918035245,
$$
which shows that the convergence radius of power series of $F_p(4)$
is $0.0918035245$.}

The proof of the first part of the fact may be found in [29]. The left side
of the inequality follows from Fact 9. Since any rooted $2$-connected
$4$-regular planar map may be {\it re-embedded} into $N_1$ by making
a twist on its root-edge (the operation is shown in Fig. 11), the set
of rooted $2$-connected $4$-regular planar maps may be viewed as an infinite subset of ${\cal U}_p(4)$,
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the set of rooted 2-connected
4-regular maps on $N_1$. Thus, the rest part of the inequality follows
from Fact 8 and the table below (20).

Let $F_p(4)=(1+2z)(F_p(2)-1)+3F$ be as defined in Fact 5 and
$m=0.0918103549$ be the convergence radius of $F_p(4)$. In order to
apply Darboux's method [2, Theorem 4], we should study the behavior
of $x=x(z)$ near $m$. Since both
$\frac{{\partial}^2H(x,z)}{\partial x^2}$ and
$\frac{\partial H(x,z)}{\partial x}$ are not equal to zero (where the
function $H(x,z)$ is defined by the Lagrangian inversion formula
in (17)), $x=x(z)$ may be expanded into a power series of
$(1-\frac{z}{m})^{\frac{1}{2}}$ near $m$. Let
\begin{align}
x&=a+b\bigg(1-\frac{z}{m}\bigg)^{\frac{1}{2}}+c\bigg(1-\frac{z}{m}\bigg)+
d\bigg(1-\frac{z}{m}\bigg)^{\frac{3}{2}},\nonumber\\
z&=m-m\bigg(1-\frac{z}{m}\bigg).
\end{align}
Substitute those into the equation system (18) and let the coefficients
of $(1-\frac{z}{m})$ and $(1-\frac{z}{m})^{\frac{3}{2}}$ be zero.
Further, by using a polynomial $24a^2-28a-9$ induced in (20) to reduce
the orders of the corresponding polynomials appearing the identities
obtained, we may find a group of relations such as
\begin{align*}
a&=\frac{7+\sqrt{103}}{12},\quad b^2=\frac{\alpha}{\beta},\\
c&=\frac{m(8a^3m-12mb^2a-2a^3+5a^2+6b^2a-6a-5b^2)}
{-6m+12m^2a^2-6ma^2+10ma+3},\\
d&=\frac{b(3a+5a^2m-4a^3m+4a^3m^2+5b^2m-6ab^2m+12am^2b^2)}
{a(-6m+12m^2a^2-6ma^2+10ma+3)},
\end{align*}
where
\begin{align*}
\alpha&=(1+2m)(-559498752am+3077533008am^2+40310784a\\
&\quad+50868841072am^6-63747912240am^5-10902385800am^3\\
&\quad+26896643452am^4+692327412m^2-81041472m+7057741014m^4\\
&\quad-2826562311m^3-16728212988m^5+13348368792m^6),\\
\beta&=432m(279410720am^6+265248am-23432500am^3\\
&\quad-129887248am^5+172029608am^4+22625424am^2\\
&\quad+279936a+45176364m^4+1242216m-5852502m^3\\
&\quad-155520+4298103m^2+73304640m^6-33897240m^5).
\end{align*}

\noindent{\bf Remark.}\quad We only need to expand $x(z)$ into the form
of (21). If necessary, higher terms have to be introduced and hence
more parameters such as $a, b, c, d$ will appear.

Now, we begin to investigate the asymptotic behavior of $F_p(2)$
near $m$. By (16)
$$
1-vx=2\sqrt{1-4x^2z}-1,
$$
which implies that
$$
\sqrt{1-vx}\approx
\sqrt{\frac{-8amb}{\sqrt{1-4a^2m}}}{\left(1-\frac{z}{m}\right)}^{\frac{1}{4}},
$$
and therefore
$$
\frac{1-\sqrt{1-vx}}{xz(2f-1)}\approx -0.5079632059
{\left(1-\frac{z}{m}\right)}^{\frac{1}{4}}.
$$
By Darboux's method, the coefficient of $z^n$ of the power series of
$F_p(2)$ is asymptotic to
$$
-0.5079632059\frac{n^{-\frac{5}{4}}m^n}{\Gamma(-\frac{1}{4})}.
$$


This together with the approximate values of the coefficients
of $F$ obtained in [29] shows the following

\noindent{\bf Theorem D.}\quad {\it The number of rooted 2-connected 4-regular
maps on $N_1$ with $n$ inner faces is asymptotic to
$$
-\frac{C_1(1+2m)}{n^{\frac{5}{4}}m^n\Gamma(-\frac{1}{4})},
$$
where $\Gamma(x)$ is the gamma function and $m=0.0918103549,\,
C_1=0.5079632059$.}

Finally, we evaluate the asymptotic values of the number of rooted
2-connected 4-regular maps without loops on $N_1$. All the discussions
next are based on the conditions that $y=1$ and $t=w=0$. Furthermore,
we still let $x^2=x=\eta$ denote a double root of (11) in convenience.
Now expressions of (17) and the function $F$ may be rewritten as
\begin{align}
x&=1+zx^2\left\{\frac{(1+x)^2}{2(1-x^2z)}-2\right\}=1+2x^2zf(f-1),\nonumber\\
F&=\frac{1-x-2xz+2x^3z(1+z)}{2x(1-x^2z)},
\end{align}
where $f$ satisfies (16).

As we have reasoned previously, we should first locate the singularity
which corresponds to the convergence radius of the function $F_p(4)$.
Theorem C guarantee the following

\noindent{\bf Fact 12.}\quad{\it The singularities of $F_p(4)$ satisfy either\\
\noindent $(1)$\quad $2f-1=0$, \\
\noindent $(2)$ or\quad $x=0$, \\
\noindent $(3)$ or\quad $vx=1$, \\
\noindent $(4)$ or\quad Equations in $(22)$.}

After a very similar procedure as we did in our proof of Fact 11, the
singularity determined by the sources in Fact 12 which corresponds to
the convergence radius of $F_p(4)$ is located at $\frac{27}{196}$, i.e.,
both of the enumerating functions of rooted 2-connected 4-regular maps
without loops on the sphere and the projective plane have the same
convergence radius. But as we will see that their asymptotic behaviors
near $\frac{27}{196}$ are basically distinct. Also we notice that nearly
all the discussions used in our proof of Theorem D are still valid except
for the corresponding parameters such as $a, b, c, d$ and $m$, we
obtain the following

\noindent{\bf Theorem E.}\quad {\it The number of rooted 2-connected 4-regular
maps without loops on the projective plane with $n$ inner faces is
asymptotic to
$$
-\frac{C_3}{n^{\frac{5}{4}}m^n\Gamma(-\frac{1}{4})},
$$
where
$$
C_3=\frac{\sqrt{\frac{8abm}{\sqrt{1-4a^2m}}}}
{am\left(1+\frac{-1+\sqrt{1-4a^2m}}{a^2m}\right)},
$$
and
$$
a=\frac{7}{6},\quad m=\frac{27}{196},\quad
b=\frac{7\sqrt{18766662918506}}{61089624}.
$$}

\noindent {\bf Acknowledgements}\quad The computation of the approximate resultants were carried with the
aid of algebra system (MAPLE). The authors would like one of referees who's suggestion to ignore the effects of
multi-edges greatly simplifies the case analysis and many of the expressions and consequently improves the presentation
of the paper.

\baselineskip 12pt


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\bibitem{[3]}E.A. Bender, L.B. Richmond, A survey of the asymptotic
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