On the number of trees having k edges in common with a graph of bounded degrees

DISCRETE MATHEMATICS ELSEVIER

Discrete Mathematics 169 (1997) 283-286

Note

On the number of trees having k edges in common with a graph of bounded degrees Ioan Tomescu* Faculty of Mathematics, University of Bucharest, Str. Academiei, 14, R-70109 Bucharest, Romania Received 1 December 1994; revised 26 March 1996

Abstract

In this paper it is proved that for any fixed r and k the number T(G; n, k) of spanning trees of K, having k edges in common with a fixed subgraph G of K, having s edges and degrees not larger than r, satisfies lim,~ ~ T(G; n, k)/n"-2 = (22)k e-2~/k! where 2 = lim,~ ~osin. This solves a conjecture raised in Tomescu (1985).

1. M a i n result

Let G be a s u b g r a p h with s edges e l , ... ,es of K , . F o r a n y selection K of i edges of G we shall d e n o t e p(K) = 0 if edges of K induce cycles a n d p(K) = [I~= 1 mj if these edges s p a n a forest FK of K . consisting of q = n - i trees t h a t contain, respectively, ml, ... ,mq vertices (ml + .-- + mq --- n). It is well k n o w n (see [1]) t h a t the n u m b e r T(FK) of s p a n n i n g trees of K . that c o n t a i n FK is given by

T(FK) -- p ( K ) n u- z.

(1)

In o r d e r to e v a l u a t e the n u m b e r of s p a n n i n g trees of K , having exactly k edges in c o m m o n with G of b o u n d e d degree we need two p r e l i m i n a r y results. L e m m a 1.1. Let r and i be fixed natural numbers and G be a graph with s edges such that

d(x) <~ r Jbr every x ~ V(G). I f Mi(G) denotes the number of matchings of G containing i edges, then lim

Mi(G) si

1 i! "

* E-mail: [email protected].

0012-365X/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PI1 S 0 0 1 2 - 3 6 5 X ( 9 6 ) 0 0 1 1 1-2

(2)

I. Tomescu/ Discrete Mathematics 169 (1997) 283-286

284

Proof. An edge ej, of G can be chosen in s ways; another edge ej~ having no c o m m o n extremity with ejl can be chosen in at least s - (2r - 1) ways, and so on. It follows that the n u m b e r of ordered selections of i pairwise nonadjacent edges of G is greater than or equal to

s(s - (2r -- 1))-.. (s -- (i - 1)(2r - 1)). Hence,

1

s(s - (2r - 1)).-. (s - (i - 1)(2r - 1)) ~< Mi(G) <~(~i)

which implies (2).

[]

L e m m a 1.2. Let G be a subgraph of K . such that m a x ~ via) de(x) <. r and T be a tree with q edges (1 <~ q ~< n - 1). The number of subtrees of G that are isomorphic to T is less than or equal to nr q. Proof. Suppose V ( T ) = {x~, ... ,xq+ 1}. Vertex xl can be chosen a m o n g vertices of G in I V(G)I ~< n ways; the vertices adjacent with x~ can be chosen in at most r dT~l~ ways; if y is adjacent with xl and dr(y) >~ 2 then its neighbors different from x~ can be chosen in at most r tiT(y)- 1 ways a m o n g vertices of G and so on. It follows that the n u m b e r of subtrees of G that are isomorphic to T is less than or equal to q+l

nrZ,= l (d(x,)- 1)+ 1

=

nrL

[]

Theorem 1.3. For any fixed k and r let T (G ; n, k) denote the number of spannin 9 trees of K . havino k edges in common with a fixed suboraph G with s edges of Kn such that maxx ~ vt~l de(x) <~ r and lim.~ ~o sin = 2 ~ [0, r/2]. Then the followin9 relation holds: ,-~lim T ( G ; n , k ) / n "-2 = (22)k 2 ; ~ ke !-

(3)

Proof. Let Ai denote the set of all spanning trees of K , containing edge el of G for 1 ~< i ~< s. Since T(G; n, k) is the n u m b e r of trees which belong to precisely k sets Ai by J o r d a n ' s sieve formula, we deduce

i=k

K c_ {1 ..... s}

i=k

K ~-- E(G)

IKI=i

j

p(K)n . - I - 2 IKl=i

I. T o m e s c u / D i s c r e t e M a t h e m a t i c s 169 (1997) 2 8 3 - 2 8 6

285

by (1). We can write i

p(K) = ~ p(i,m), K ~ E(G)

(4)

m~ 1

I,g] =~

where p(i,m)= ~r p(K) and the last sum is over all selections K ~_ E(G) such that IKI = i and the n u m b e r of components having at least two vertices that are induced by K in K . is equal to m. We have p(i, i) = 21M~(G), hence by L e m m a 1.1 it follows that lim p(i, i)n"-i-~ = (22) I F/n-2 i!

n~ac

It is clear that ifJKJ = i then p(K) ~< 2 i, hence for m < i we can write, by L e m m a 1.2 that

p(i,m) <~2in"ri~(i,m),

(5)

where ct(i,m) denotes the number of pairwise nonisomorphic forests with i edges consisting of n trees having each at least two vertices. F r o m (5) it follows that for every fixed i

p(i,m)n "-i-z g/n-2

~< (2r)ie(i, m)n'-i,

hence l i m , ~ p(i,m)n"-i-2/n "-2 = 0 for every 1 ~< m ~< i - 1. N o w (4) implies that lira ~ p(K)n"-i-a/n "-2 (22)i , ~ o K~_e(~l = i!

(6)

IKI = i

F r o m the general theory of the principle of inclusion and exclusion, it is well known that the alternating sum defining T(G; n, k) satisfies the Bonferroni inequalities (see e.g. [2-]) and by standard methods from (6) we deduce that lira "~

n,_ 2

and (3) is proved.

=

(-1)' i=0

i+k k

(22)'+_~k -2a (i + k)! = k----~ e

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This result implies that the r a n d o m variable taking the value k with the probability

T(G; n, k)/n"-2 is distributed in accordance with the Poisson law with parameter 22 as n --, oc whenever l i m , . ~ sin -- 2 exists. This was proved in [3] only in the case when G is a caterpillar with s edges of b o u n d e d degrees and it was claimed that (3) still holds for general trees of bounded degrees.

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L Tomescu / Discrete Mathematics 169 (1997) 283-286

References [13 C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1970). I-2] E.M. Palmer, Graphical Evolution: An Introduction to the Theory of Random Graphs (Wiley, New York, 1985). 1-3] I. Tomescu, On the number of trees having k edges in common with a caterpillar of moderate degrees, Ann. Discrete Math. 28 (1985) 305-310.