AbstractThis paper deals with the enumeration of rooted planar maps in which the root vertex is of arbitrary valence and all other vertices are trivalent. A formula, in explicit form, is given and closed form expressions are given for several cases of interest. Some interesting summation formulae are also obtained towards the end of the paper

Mathai, A.M

Rathie, P.N

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JOURNAL OF COMBINATORIAL THEORY (B) 13, 83-90 (1972) 
Enumeration of Almost Cubic Maps 
A. M. MATHAI AND P. N. RATHIE 
Department of Mathematics, McGill University, Montreal, Quebec, Canada 
Communicated by W. T. Tutte 
Received July 20, 1971 
This paper deals with the enumeration of rooted planar maps in which the 
root vertex is of arbitrary valence and all other vertices are trivalent. A formula, 
in explicit form, is given and closed form expressions are given for several 
cases of interest. Some interesting summation formulae are also obtained 
towards the end of the paper. 
1. INTRODUCTION 
The aim of this article is the enumeration of rooted trivalent planar 
maps in which the root vertex has arbitrary valence and all other vertices 
are trivalent. This is done by considering planar maps with the root face of 
arbitrary valence and all other faces of valence 3. Tutte [4] has solved 
the problem of rooted planar maps with root face a k-gon and all other 
faces p-gons (p-even). When p is odd the only cases solved are p = 3, 
k = 1,2, and 3 by Mullin [2]. Mullin, Nemeth, and Schellenberg [3] 
considered the case p = 3 for all k and obtained numerical tables for the 
number of admissible maps of the type (n, m) for IZ, m = 1,2,..., 14. An 
admissible map is a rooted planar map with root face of arbitrary valence 
and all other faces trivalent and it is of type (n, m) if it has n interior 
faces and exterior faces of valence m. We follow Tutte [5] for the definitions 
of planar map, face, root, valence, etc. The explicit expression given in [3] 
for the number of admissible maps of the type (n, m) does not seem to 
contain sufficient number of conditions for the validity of the results. 
In this article we obtain the explicit expression for the number of admissible 
’ maps of the type (n, m) and a number of other interesting results. 
83 
Copyright 0 1972 by Academic Press, Inc. 
All rights of reproduction in any form reserved. 
58zb/q/r-6 
84 MATHAI AND RATHIE 
2. DETERMINATION OF t,,,, 
Let t,,, be the number of admissible maps of type (n, ~2). Then the 
formal series, 
T(x, J) = f f tn,mX’yn, (1) 
n=o na=O 
is the generating function for these numbers. A functional equation for T 
is given below, 
T= 1 +y2T2+ x+(T-~L - I), (2) 
where 
L = L(x) = f tn,lxn (3) 
a=0 
is the generating function for maps whose root face is a loop. The 
functional equation (2) can be easily obtained from [5, p. 691. 
Equation (2) may be rewritten as 
T = [I - xy( y - x)-l L] + y”( y - x)-l T2, (4) 
which on applying Lagrange’s expansion [6, p. 1321 gives 
T= f 
(2n) ! 
n=o n! (n + l)! 
[ y”( y - x)-l]fl [l - xy( y - x)-l L]“il. (5) 
Now expanding the terms inside the sum by using binomial expansion 
and keeping in mind that the powers of x and y are non-negative, we get 
m  n+1 an 
+ go f. n, (12’“:! 1), (” + j - ‘) x!Y-j. (6) 
ENUMERATION OF ALMOST CUBIC MAPS 85 
An expression for (xL)k is given in [3] that can lead to wrong values for 
t n,m . The correct expression in a simplified form is given below: 
m min(k-l.m-k) m-k-i 
(XL)” = c 
m=k 
z. go (+)r ; ')(* + 4" - 1)(-3)"22m-2ri-g 
x 
for m>k>l. (7) 
The binomial coefficient (:) is written with the convention (y) = 0, when 
r > n. 
Thus (6) with the help of (7) takes the following form: 
m 2n n+1 co 03 2n 
T = C 2 C C f(n, k, j, m) xZm+jy2+i + c c f(n, 0, j, 0) x*y2’+j, 
n=O j=O k=l m=k n=O j=o 
(8) 
where 
f(n, k, j, m) = n, (yi!,t! m (” z ‘)(” ’ k T j - ‘) (-l)k 
min(k-l,m-k) m-k-i 
X z. C ("; ')(" +4" - ') (-3)" 22m-Sk-i-g 
4=0 
x 
[( )( 
; “mm-i ;“; 2) - ($)(“,-_;’ ; “)I, 
and 
f(n,O,j,O)= n,$n:!l),~+~-l). 
It is easy to see by actual expansion of the sums that 
n+1 m 
zl mzkf(n, k, j, m) x2” = g (I$ + mz+jf(n3 k, jY m) x2m 
(9) 
(10) 
01) 
= (F1 $’ + y1 i 
m=l k-l k=lm=n+2 
) f(n, k, j, m) x2”. 
86 MATHAI AND RATHIE 
Hence (8) and (11) yield 
T = f  T i nilf(n, k, j, m) .fhii4,2n+ 
n=O j=O m=n+2 k=l 
co 2n n+1 m 
+ c 1 c 1 f(n, k, j, m) x2@jyzn-j 
n=O j=O m=lk=l 
+ f E f(n, 0, j, 0) xiyzn--j, 
n=o j=o 
(12) 
wheref(n, k, j, m) andf(n, 0, j, 0) are given by (9) and (lo), respectively. 
The third term in (12) can be absorbed in the second, if necessary, with 
the convention that, when k = 0, m = 0 and k/m = 1. 
Now comparing the coefficients of xzSy2+ and x2s+1y27-1 for r, s = 0, I,... 
separately in (12), we obtain 
[&+r-2)1 7+YP+1 
t 2s,2r = z. zl f(r + P, k, 2~~ s - P) 
S-l S-D 
-I- c 1 f(r + P, k, ZP, 3 - P) + f(r + s, 0, 2s, 0), (13) 
a=[+(~-r)] k=l 
and 
t 2s+1*2r--1 
[j(s-r-211 r+r+1 
= z. & f(r+p,k2p+L~-P) 
S-l s--P 
+ p-,L~p,,, ,&fk + Pi k, 2~ + 1, s -P) +f(r + s, 0,2s + &O), 
I 
(14) 
where the convention [x] = greatest integer in x < x is followed and 
f’s are given by (9) and (IO). Also, when [$(s - r)] is negative it is inter- 
preted as zero and the usual convention that & (a) = 0, if b < a, is 
followed. 
When s = 0 in (13) and (14), only the last term, namely,f(r + s, 0,2s, 0) 
and f(r + s, 0, 2s + 1, 0), respectively, will exist which will readily give 
t o.2+ and t1.21-l . When r > s - 2, then in (13) and (14) there will be no 
contribution from the first term on the right-hand side of each. 
ENUMERATION OF ALMOST CUBIC MAPS 87 
3. SOME INTERESTING PARTICULAR CAKES 
In this section, we will derive closed form expressions for t,,, in several 
cases 
(i) Since the right-hand side of (12) gives the coefficients of xpy* only 
when p + q is even, 
t 9.9 = 0, 
whenever p + q is odd. 
(ii) Writing (2) in the form, 
y3T2+(X-y)T-xyL-x+y=o 
and substituting the expression for T from (1) in (16) we have 
(15) 
(16) 
2 2 [( f f tr, sLm-,) xnym+3  tn.mXn+lym - tn.mx~y”+‘] 
n-o m=o s=o r=o 
-i. 
tn,lXn+ly - x + Y = 0. (17) 
Now, comparing the coefficients of xn+lyo in (17), we get 
t - 0, a.0 (18) 
for n > 1. 
(iii) Taking s = 0, 1 and 2, respectively, in (13) and (14) and utilizing 
(9) and (lo), we have 
t 0.2+ = cwW(~ + l)!), r = 0, 1, 2 ,... . (19) 
t1.2r-l = (2r) !I{@ - I)! (r + 99 = rt0.27 , r = 1, 2,... . (20) 
t - 2(2r)!/{(r - l)! r!} = 2r(r + 1) to,2r, 2.2r - r = 1, 2,... . (21) 
t3.2r-1 = 2(r + 2)(2r)W(r - l)! r!> = (2/3) r(r + l)(r + 2) t0,2r , 
r = 1, 2,... . (22) 
t - 2(r2 + 7r + 16)(2r)!/{3(r - l)! r!} 4.2r - (23) 
= CW r(r + l)(r2 + 7r + 16) t0.2r , r = 1, 2,... . 
t5,27-l = 2(r3 + llr2 + 50r + 58)(2r)!/{15(r - l)! r!} (24) 
= (2/15) r(r + l)(r3 + llr2 + 50r + 58) t0,27, r = 1, 2,... . 
88 MATHAI AND RATHIE 
From (19) to (24), we can conjecture that 
t 2n,2m = p2nt0,2m 
and 
fzn-1,2m-1 = Q2n&,2m 9 
(25) 
(26) 
where Pz, and QSnP1 are polynomials in m of degree 2n and 2n - 1, 
respectively. 
(iv) Mullin [2] has shown that 
tzs+l,l = 23s~W2 + I)/@ + l)! &/2 + 2)), s = 0, 1) 2 )... . 
(27) 
Comparing the coefficients of x~~+~J+? and x2sy3 in (17), we get 
b-l,3 = t2s.2 = t2s+l.l > (28) 
for s > 1. 
The relation t2s-l,3 = 22S+1,1 can also be proved from (14). For that, 
taking I = 1 in (14) gives an expression for t2s+l,l . Now replacing s 
bys--l1,rbyr+1,jbyj+2andthenputtingr=1in(14)weget 
an expression for t2S-1,3 . We can account for the combination of n, k, j, m 
with j = 0 by putting p = - 1 in the sum. It is easily seen from these 
two expressions that t2s--1,3 = t2s+l,l . 
Again, comparing the coefficients of xZ9v4 in (17) and utilizing (28), 
we have 
t -t 29,4 - 2s+3.1 - 2tzs+1,1 * (29) 
Now t2s.4 , t2s-1.3 and t2s,2 can be put into closed forms by using (27). 
In a similar manner, more coefficients can be put in closed forms. In 
n general, comparing the coefficients of x y m+3 in (17), we have the following 
recurrence relation 
(30) 
for n > 0, m >, 0. When n = 0, the term tn-l,m+3 in (30) is taken as zero. 
The recurrence relation (30) can also be used for computing t,,, for 
various values of n and m. 
4. SUMMATION FORMULAE 
This section deals with some interesting summation formulae obtained 
either from the recurrence relation (30) or by equating the two types of 
expressions obtained for the same coefficient in T. 
ENUMERATION OF ALMOST CUBIC MAPS 89 
(i) The equation (30) for n = 0 with the help of (15) and (19) gives 
5 (2s)! (2m - 2s)! (2m + 2)! s,os! (s + l)! (m - s)! (m - s + l)! = (m + l)! (m + 2)! ’ (31) 
(ii) Taking n = 1, m = 2r - 1 in (30) and utilizing (15), (19) and (20), 
we have 
k-l 
c 
(2s) ! (2k - 2s) ! k(2k + 2) ! 
s=o s! (s + l)! (k - S - l)! (k - S + l)! = 2(k + l)! (k + 2)! * (32) 
More results of the type (31) and (32) can be obtained from (30) by 
taking n = 2, 3,... and using equations (19) onwards. 
(iii) The equations (13) and (14) for r = 1, (28) (9) (10) and the result 
(3 = (3 + L+nl), give 
[[fy z2 + 
Z-0 k=l 
y] (p + l;~;wY(s _ p) 
P=[+(s-l)] k=l 
X (’ ; ‘)(; ; :, (-I)“+1 min’k~-9-k’ ‘-g+ (” ; ‘) 
(-3>” 22s-2p-2k--i-q 
x 3 2s-2p-i-q-2 
[( I( z s-pfk-2 ) 
(33) 
(iv) Lastly, we will show that, 
m-1 
c 22+-lt -t 2,2m-2T - 3,2m-1 - f2.2nz 3 (34) 
r=l 
or, alternatively, 
m-1 
~~22Y2m-2r)!/{(m-r-l)!(m-r)!} 
= 2(2m)! / (3(m - 2)! m!}. (35) 
The left-hand side of (35) on using n! = F(n + l), the duplication 
formula for gamma functions [I, p. 5 (15)] and [I, p. 104 (46), p. 52 (6)] 
proves the desired result on interpreting with the help of (21) and (22). 
90 MATHAI AND RATHIE 
ACKNOWLEDGMENT 
The authors would like to thank Professor R. C. Mullin for introducing them to 
enumeration problems in graph theory as well as for some useful discussions. 
REFERENCES 
1. A. ERDBLYI et al., “Higher Transcendental Functions,” Vol. I, McGraw-Hill, 
New York, 1953. 
2. R. C. MULLIN, On the average number of trees in certain maps, Canad. J. Math. 
18 (1966), 33-41. 
3. R. C. MULLIN, E. NEMETH, AND P. J. SCHELLENBERG, The enumeration of almost 
cubic maps. 
4. W. T. TIE, A census of planar maps, Canad. J. Math. 15 (1963), 249-271. 
5. W. T. Turre, On the enumeration of planar maps, Bull. Amer. Math. Sm. 74 (1968), 
64-74. 
6. E. T. WHITTAKER AND G. N. WATSON, “A Course of modern Analysis,” Cambridge 
University Press, Cambridge, 1962. 


Enumeration of almost cubic maps 

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Enumeration of almost cubic maps

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