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Matchings in Polytopal Graphs

6. Gfinbwm University of Washington Seattle, Washington

1.

A s e t M of edges of a graph G i s a matching (or independent)

A matching set M i s a matching of G pro- set of G provided each vertex of G i s incident with a t most one edge t h a t belongs to M. vided there is no matching s e t of G which properly contains M. The notion of matching should be distinguished from the more commonly encountered "max imum matching"; the l a t t e r denotes (see, fo r example, O r e 117, Ch. 71, Harary [ll, Ch. 101 , Berge [ 3 , Ch. 73) a matching with the l a rges t possible number of elements.

The following pages are devoted t o an investigation of some propert ies of matchings, i n par t icu lar those of planar and poly- topal graphs. The r e su l t s obtained are ra ther simple, and I believe t h a t they would have been discovered long ago i f it had not been for the a l l too frequent preoccupation with maximum matchings. ( In order to appreciate the handicap imposed by the "maximum" re s t r i c t ion , the reader should t r y t o imagine the theory of connectivity i n graphs i n which the only cut-sets con- sidered are those having cardinal i ty equal t o the degree of connectedness.) I t is cer ta in t h a t a s i m i l a r "non-discriminatory" approach should lead t o a bevy of new re su l t s on coverings and i n many other areas of graph theory.

f e w simple observations to provide a backdrop for our resu l t s .

ver t ices and edges of G. If M is a matching of G we denote by e(M) the number of edges i n M; by n ( G ) and m(G) we sha l l denote the least and la rges t numbers of edges i n a matching of G.

(compare Berge [2], Norman-Rabin 1161 ) t h a t fo r every graph G and every integer m with m(G) 2 m 5 m ( G ) there ex i s t s a matching M of G such t h a t e ( M ) = m. I t is easy t o ver i fy t h a t f o r every graph G w e have m(G) 5 2 g ( G )

We begin by introducing some notation and by formulating a

For a graph G we denote by v(G) and e(G) the numbers of

Using the technique of "alternating paths" it follows eas i ly

Networks, 4: 175-190 @ 1974 by John Wiley & Sons, Inc. 175

Clearly 6 ( G ) < v(G) /2; M i s a matching of G such t h a t e ( M ) = v(G)/2 i f and only i f M is a 1-factor of G. The quantity ( v ( G ) - s ( G ) ) / i ( G ) may be used as an indicat ion of the measure i n which G f a i l s t o have a 1-factor. E s t i m a t e s of t h i s quantity i n terms of other parameters w e r e given by Weinstein [211 and Gallai [71.

L e t K(p ,p , . . . ,p ,) denote the complete j -par t i te graph 1 2 3 . . . , p . elements, the

3 i with sets of ver t ices containing p 1' P2,

notation being such t h a t p 1 9 , - < ... - < p j , and l e t p = Then it is eas i ly proved t h a t

pi. i=l

i(K(PlrPZr**-.P 1) = d n { [P/2I r P - Pj} j

(see Chartrand-Geller-Hedetniemi [4 , p.331); moreover it may be ver i f ied t h a t

g(K(P1,p2t-.-rp.)) = m I P j - l i I (P - Pj ) /2 [ I= 3

(Here ]x[ denotes the s m a l l e s t in teger not less than x.) Then

v(Id) = 2 and e(Id) = d2d-1; using the matching indicated by

the heavily drawn edges i n F igure 1 it is easy t o see t h a t g(13) = 3.

Let Id denote the graph of the d-dimensional cube.

d

Forcade [6] proved t h a t g( Id) /v( Id) i s a non-

increasing function of d , and t h a t l i m g ( Id) /v( Id) = 1/3. d++

Fig. 1

The re la t ions between g ( G ) , m(G) , v(G) and the connectivity of the graph G appear t o be worth investigating. As sample re- s u l t s we may mention:

M A T C H I N G I N POLYTOPAL GRAPHS 177

There exis t j-connected graphs G with a r b i t r a r i l y large v ( G ) such that m(G) = [(j+1)/21 (compare Figure 2 for j = 3) ; c lear ly g ( G ) 1. [(j+1)/21 fo r every j-connected graph G. On the other hand ;(GI 2 j f o r every j-connected graph G with suf f i - c ien t ly many ver t ices , and equality is possible f o r a r b i t r a r i l y la rge v(G) (see Figure 3 f o r j = 3; j-connected graphs of t h i s type s a t i s f y ~ ( G I = ;(GI = j ) .

Fig. 2

Fig. 3

A subfamily of the d-connected graphs is formed by the d-pozytopd graphs, t h a t i s , graphs isomorphic to the graphs of edges and ver t ices of d-dimensional convex polytopes. I t i s a w e l l known theorem of S te in i t z [201 t h a t a graph G i s 3-polytopal i f and only i f it is 3-connected and planar, but no character- i za t ion of d-polytopal graphs is known. polytopes and on polytopal graphs, and for references to the l i t e r a t u r e , see GrAinbaum [81 , 191 1 .

One of the areas of our i n t e re s t i s the investigation of upper and lower bounds for ;(GI and = ( G I , when G is permitted t o range over a l l d-polytopal graphs with v ver t ices , v > d > 2.

the t r i v i a l estimate & ( G ) 5 v ( G ) / 2 fo r d-polytopal graphs G with a r b i t r a r i l y large d and v(G).

(For r e su l t s on convex

The example of d-zonotopes shows t h a t equality can hold-in

L e s s t r i v i a l are the r e s u l t s concerning

g(v,d) = min {m(G) I G is d-polytopal and v(G) = v}, g,(v,d) = min f m ( G ) I G i s d-polytopal and v(G) = v). and

W e have

Theorem 1: such that:

There &st posi t ive constants c: a d c?, d - > 2,

In case d = 3 w e have the following more precise r e su l t :

Theorem 2: vertices, and l e t M be a matching i n G wi th m = e(M) edges. Then 6m zv + 4, and equality is possible for a l l values of m > 2.

Let G be a 3-connected p Z m graph with v = v ( G )

- Another group of r e s u l t s concerns packings of matchings. ;(GI denote the l a rges t number of matchings of G such t h a t L e t

no edge belongs to more than one of them and l e t IT(G) denote the l e a s t number of edge-disjoint matchings of G such t h a t every other matching of G has a t l e a s t one edge i n common with one of t h e m .

W e have

Theorem 3: graphs G i s 12.

The largest value of G(G) possible for 3-polytopal

Denoting by K the complete graph with v ver t i ce s , it is eas i ly checked t h a t ( f o r n 2 2) and ;(K2n+l) = 2n+l.

- V (K2n) = q ( K p n ) = ;(Kzn+l) = 2n-1,

Since it is w e l l known t h a t K i s V

d-polytopal whenever v > d > 4 , it follows t h a t no generalization of Theorem 3 t o d-polytopalgraphs i s possible for d > 4.

d-polytopal graphs G we have Concerning the minimal values of ;(G) possible f o r

Theorem 4 : there exist d-polytopal graphs G with v(G) = v such tha t ,(GI - .c 3.

For every d 2 2 and for every su f f i c i en t l y large v

This is complemented by

Conjecture 1 : I f G is d-polytopal and d - > 3 then ;(GI - > 3.

MATCHING I N POLYTOPAL GRAPHS 179

Regarding 9(G) we have

Theorem 5: there ex i s t d-poZytopa1 graphs G with v ( G ) = v such that - r ( G ) 2.

For every d 5 2 and for a l l su f f ic ien t ly large v

This r e s u l t is complemented by

Conjecture 2: If G i s d-polytopal and d '> 2 then Z(G) 1. 2.

The proofs of the above theorems are presented i n Section 2 . are collected and discussed i n Section 3 .

Various additional remarks, open problems and conjectures

2. P W F S

We give f i r s t the proof of Theorem 1.

L e t p(v,d) denote the m a x i m a l number of face ts ( t h a t i s , (d-1)-dimensional faces) possible f o r a d-polytope with v ver t ices . It is w e l l known (see K l e e 1121, McMullen 1151) t h a t

(x ( v-fd) fd fo r even d

so t h a t

[Wl + k v d

where kd = ~ ~ / ( [ I q d l l ) and cd = 1 o r 2 depending on whether d is

even or odd. p ( t , d ) face ts ; cycl ic polytopes provide examples of such poly- topes. W e construct a d-polytope C*(t,d) by choosing f o r each face t of C ( t , d ) a point outside C(t ,d) but near the centroid of that facet, and taking the convex hu l l of the union of C(t ,d) and the set of the p ( t , d ) "new" points. Then the "new" ver t ices of C*(t,d) are connected exclusively to "old" ver t ices t h a t belong t o C( t ,d ) . Therefore each edge of every matching of C*(t ,d) contains a t least one of the t ver t ices of C( t , d ) , and so the matching contains a t most t edges. On the other hand,

L e t C(t ,d) be a d-polytope with t ve r t i ce s and

Consequently t c:* v~"'~' , and the proof of one of the

inequal i t ies is completed.

I n order t o prove the other inequality ( for d 2 4 , since the case d = 2 is obvious, and d = 3 is t reated separately i n Theorem 2 ) we r e c a l l the following theorem of Klee [131:

ver t ices , n > d+l , has a t most p(n,d) connected components.

v ver t ices be given. Then, by omitting the 2m ver t ices of M from G we obtain a t most v(2m,d) connected components; but each of these components i s a s ing le vertex since M i s a matching.

The graph obtained from a d-polytopal graph by omitting n

Let a matching M with m edges i n a d-polytopal graph G with

[#dl m[3d1 + 0 (mr3dl-l) , so that Thus v 5 2m + v(2m,d) 5 kd 2 , as claimed. T h i s completes the proof 1/ [#dl m 2 g,(v,d) 2 c i v

of Theorem 1.

Turning now to the proof of Theorem 2 , we s h a l l f i r s t show t h a t for each value of m the possible values of v a re bounded from above; then we s h a l l determine the s t ruc ture of the graphs with m a x i m a l v, and from t h i s derive the inequality.

Let G be a 3-connected planar graph with v ver t ices and a matching M with m edges. vertices t h a t contains G and has M a s a matching, such t h a t G* t r iangulates the plane. successively adding edges: more s ides , a t least 2 non-adjacent ver t ices V V of Q belong

t o M; then we may adjoin the edge (V V 1 without disturbing M

as a matching. Observe t h a t (V V ) i s not an edge of G since

i f it were G would not be 3-connected.

least 2 ver t ices of each face of G belong to M, and i f a vertex is not i n M then a l l i t s neighbors are i n M. from G a multigraph G i n the following manner: For each edge

Then there e x i s t s a graph G" with v

Indeed, G* may be obtained from G by I f G contains a face Q with 4 o r

1' 2

1' 2

1' 2

Thus we may assume t h a t G i s a tr iangulation; c lear ly a t

W e now construct A

A A A A A

E e M there i s i n G a vertex E ; two ver t ices E, and E, of G a re I L

joined by an edge such t h a t F fl El # pl # F n E2. such edges F fo r a given p a i r E E w e see t h a t G may have

edges with mult ipl ic i ty up t o 4. checked t h a t G is planar, and t h a t t o each vertex of G t h a t is not i n M there corresponds i n G a face. Therefore, since a planar graph with mul t ip l i c i t i e s a t most 4 and with m ver t ices has, by the Euler formula and planari ty , a t most f = 4(3m-6) - m + 2 = l l m - 22 faces, it follows t h a t the number of ver t ices of G is a t m o s t v < f + 2rn < 13m - 22.

of 6 i f and only i f there is an edge F of G Since there may be up t o four

A

1' 2 On the other hand it is eas i ly

- -

MATCHINGS I N POLYTOPAL GRAPHS 181

L e t v now be the maximal number of vertices possible i n m a 3-polytopal graph G which has a matching consisting of m edges, and l e t G

edges; we denote by M a matching of G

the reasoning used above we see tha t Gm is a tr iangulation, and

tha t a t l e a s t 2 ver t ices of each face of Gm a re i n M. However,

it is not possible t h a t a l l 3 ver t ices of a face of Gm belong t o

M, since i n tha t case we could add t o Gm another vertex, con-

nected t o j u s t those three vertices; the result ing graph would st i l l have M as a matching, contradicting the maximality of v . m Therefore, precisely 2 ver t ices of each face of G are i n M.

Another property of G we need is tha t each vertex of G m m

be such a G having the maximal possible number of

m

m containing m edges. By

m

tha t does not belong to M i s of valence 3. Indeed, assume tha t 0 is a vertex of G \ M of valence a t l e a s t 4; then its neighbors

B, A, A ' , B' are i n M (see Figure 4 ) . Let ( A , A ' ) be an edge tha t is not i n M. W e delete from Gm the edges ( A , A ' ) and ( 0 , A )

and introduce a new vertex D and edges (B,D) , (A,D) , (A'D) . Since the resul t ing planar graph s t i l l has M as a matching, the maximality of v implies t ha t the graph is not 3-connected;

therefore A ' and B belong t o the same face of G We next t r y

to delete from G the edges ( 0 , A ' ) and ( A , A ' ) , and to introduce a new vertex D and edges ( B I D ) , ( A , D ) , and ( A ' D ) . I f t h i s w e r e also impossible it would follow similarly tha t i n G

A and B' belong to the same face. But then the deletion of A and A' would disconnect G between C and 0, contradicting the

assumed 3-connectedness of G This proves our claim tha t a l l

ver t ices of G \ M must be 3-valent.

m

m

m'

m

the ver t ices m

m

m'

m The proof of the inequality is now easy: Let v be the 0

number of ver t ices i n M and v the number of vertices of G not

= v + v and v = 2m. On the i n M. Then, on the one hand other hand, deleting the ver t ices of Gm \ M from G

triangulation G* such tha t the number f

f 2 v and f 5 2v0 - 4. Therefore, v = vo + v1 L V + f o 5 3v

1 m

t v m 0 1 0 yields a m

of i t s faces s a t i s f i e s 0

0 1 0

m 0 0 - 4 = 6m - 4, as claimed. The proof of Theorem 2 is completed.

182 G ~ B A U M

Fig. 4

Fig. 5

MATCHING I N POLYTOPAL GRAPHS 183

I n order to es tab l i sh the upper bound of Theorem 3 w e r eca l l the following r e s u l t of Kotzig [141, which deserves t o be much more widely known: Every 3-polytopal graph G contains an edge E such t h a t the sum of the valences of i ts endpoints is a t most 13. For such an edge E there are a t most 1 2 match- ings of G t h a t contain a t l e a s t one endpoint of E. each matching contains a t least one endpoint of each edge this shows i ( G ) 5 12 .

shows t h a t by A, B, and the d i g i t s 0,1,...,9.) Thus Theorem 3 is completely established.

(d-2) -fold pyramid over the n-gon (n-circuit) C where

n = v - d + 2 > 8d - 15. That is, G consis ts of the n-circui t a complete graph K and a l l the edges connecting ver t ices 'n d-2'

M2, M and M are 3 4 A s s u m e t h a t M1, of C with those of K n

edge-disjoint matchings on G . Then a t most 4(d-2) ver t ices and 8(d-2) edges of Cn are incident t o edges of

M = M t h a t meet Kd - 2. Therefore, n > 8(d-2) implies t h a t there i s an edge E i n Cn t h a t i s incident to no

d-2' edge i n M t h a t has an endpoint i n K

coincide with) a t l e a s t four edges i n C

M

Cn (counting i t s e l f ) .

proved.

But since

On the other hand, the 3-polytopal graph G i n...