Volume 5, number 4 INFORMATION PROCESSING LETTERS October 1976

SOME IMPROVED BOUNDS ON THE NUMBER OF l-FACTORS OF n-CONNECTED GRAPHS +

Haroid N. GABOW Department ‘of Cbmputer Science, University (Ff Colorado, Boulder, Cola. 80309. USA

Received 2 February 1976, revised version received 23 July 1976

Graph, l-factor, matching, connectivity

1. Introduction

There is an interesting relation between connectiti- ty and the number of distinct l-factors in a graph. Beineke and Plummet [t] showed that if a graph has a I-factor and is n-connected, it has n l-factors. Zaks [3] sharpened the bound on the number of l-factors to n(n - 2)(n - 4) l l *a ’ 5*3fornoddandn(n-2) (n-4)* . . . l 4 l 2 for n even. This bound is exact for the complete graphsKn+t (n odd) aM the party grq~h Qa (rc = 4). We show these are the only graphs that achieve the bound for n 3 3, and improve the bound for the remaining cases.

b *

b

2. Pr&lniMlie!3 Q

This section gives some definitions and previous results. For terms not defined here, see [2].

A graph G that has two or more vertices is connect-

ed (I-amected) if there is a path between any ?wo vertices. For a positive integer n, G is n-connected if, when any m vertices q, 0 6 m < n, are deleted, the resulting graph C - (t+ Ii = 1, . . . , m} is connected. (Prote an naonnected graph has at least n + 1 veriiccs.j G has conwcti~&y n if it is rrconnected but not (n + t&connected. If G is connected but G -- (u) is not, u sepamtes G.

* Dept. 9f Computer Science, University of Colorado, Tech- nical Report #CU-%086-76. This work was partially sup- ported by the National Science Foundation under Grant GJ 09972.

A I-factor (perfect matching) of G is a subgraph

with exactly one edge incident to every vertex of G.

A vertex v is totally covered if eve$edge VW incident to v is in some I-factor of G. A fundamental result is the following:

Lemma !I [3]: A Z-connected graph that has a l-factor has at least two totally covered vertices.

Defme P’(G) as the number of distinct l-factors of G. For any positive integer n, define f (n) as the largest

i integer such that if G is an n-connected graph with a l-factor, t&n F(G) 2 f(n). Define g@), a variant of the factorial fun@ion, by g(n) = n for n = 1,2, and

. g(nj=ng(n&)‘forn~3.Thusg(r;b)=n(n -2)~: l 5 l 3 or n(n -- 2) l .2 4 l 2, depending on whether PI is odd or even. An induction using, Lemma 1 shows f(n) >g(n) for n 2 1.

Define f*(n) as the second-lowest possible number of l-factors for an n-connected graph. That is,/*(n) is the largest integer such that if G is an n-connected graph with F(G) >f(n), then F(Gj W*(fr).

The com:vlete graph AK, consists of n vertices and all possible edges between them. If 12 is odd 9 K,,+1 is

n-connected and has a l-factor. In fact, F&+1) = g(n); so f(n) = g(n) for n odd l

If n is even, the party graph P,, i:; &, with the edges of a l-factor deleted. Pn is (n - 2)-connectert. It is easy to check F(P4) = 2, F(P6) = 8, and F(P,& = n(F(P,) + F(P&) for n 2 6.

3. The new bounds

We begin by analyzing 3- and 4-connected graphs,

113

Vdum 5, number 4 INFORMATION PRCMXSSlNG LETTERS :’ October 1976

&nfiming conjectures of Zaks on f”(3) and f*(4). Then G - (t, v) is 2-connected, SO F(G - {t, u}) 2 2.'

Lemma 2: f(3) = 3; K4 is the only 3-connected Since t has @ agree at least four, we see .F(C) 2 8. mus

r* G with Fm = 3;f*(3) 3 4. \ * f(4)=& 4

w: &ce li(K4) = 3, we have f(3) 9; 3. Similar !y,/“(3) 6 4 is shown by the graph consisting of a cyde on ail6 vetiices, V1~2b3V,~V5V6, phS the @dgeS

ulv4, v2ve~ 93vs. Lemma I impliesf(3) Z 3, so f(3) = 3.

It redru ody to show that if G # K4 is 3 3-con-

otally covered vertex t, by jtemma 1. imply t is adjacent to exactly three

i = 1,2,3. Further, each graph 4; - {t, I.$ one lafactor, i.e., P’(G - {r, q)) = 1.

1, we see G - {t, ui) is not konnected;

is the only rlconnected

ninP+ndP~+e,wherethe

graph with a l-factor. Let x an3 let tube an edge.

It remains only to show that F(G) > 10 if G #Pa. The argument divides into two cases.

t%se I : There is a totally covered ‘vertex t and an edge tw, such that G - {t, u} is 3eonnected.

Apply the results of Lemma 2 to G - (t, u). If G - (t, u; + K4, then li(c - {t, v)l& 4, so F(G) > 4+3*2=10,asdesired.OtherwfseifG-{t,u}=&, graph. G has six vertices and contains at least the edges ofPg+e.ThusF(G)>F(P6 te)= 10.

C&se 2: For every totally covered vertex t and every edge tu, G - {t, u) has aonnectivity 2.

If a totally covered vertex has degree five or more, then F(G) 3 5 l 2 = 10, as desired, Thus assume all totally covered vertices have degree four. Assume further that F(c) < 10, Below we deduce G = Pa.

Let t be a totally covered vertex, adjacent to ver- tiU% Vi, i = 1,2,3, 4U We first Show vertices Ui form a cycle on four vertices, with no other edges joining them.

No three vertices uj form a cycle. For suppose qi19~ is a cycle, Then since G -{u4) is 3-conn?cted, C - {t, u4} is 3Gonnected. But this violates the as- sumption of Case 2.

Since P(G) < 10, assume without loss of generality that F(G - {t, q}) = 2, for i = 1,2,3. Since G - {t, vi}

is 2=connected, it has two totally covered vertices. Both have degree ‘two in G - {t, vi). kqy vertex has degree at least four in G. So both totally covered vertices of G - (t, ui} are adjacent to t and Vi* in particular, both are among the vertices ~1.

Thus each vertex ui, i = 1,2,3, is adjacent to two other vertices vi. Since no three vertices uj form a cycle, it is easy to see the four vertices vI form a cycle, with no other edges.

Now we find other totally covered vertices besides t. Without loss of generality, assume the cycle found above is v1 ~2~3~4. Vertex v2 is totaily covered in G- (t,q},fori= 1,3. Thus u2 is totally covered in G. This in turn implies vl is totally covered in G.

The above. argument shows the totally covered ver- tex u2 is adjacent to four ve~tirss joined in a cycle. This cycle is ultu3s, where s is a vertex in G. Similarly, u1 is adjacent to four vertices joined in a cycle, v2 tq~. l’%us the six vertices s, t, y, i * 1,2,3,4, and their interconnecting edges, form the party graph Pe l

Volume 5, num’ber n4 INFORMATION PROCESSING LETTERS October 1976

G contains no other vertices, since G - {s, 9, ~4)

is connected. Thus G = &. QED 0 Now we extend these results to higher connected

graphs. Zalcs conjectured that Km,1 (n odd) and fG (PI = 4) are the only graphs with exactly&) I-fastors, for 3. Theorems 4 and 5 confirm this.

eorem 4: Let n > 3 be odd. Then f(n) =g(n); K n+l is the y n-connected graph C with F(G) =

g(n); f’(n) s(n)* Roof: We show by induction on odd n that if

G#K,+l is an n-connected graph with a I-factor, then F(G) 3 $ g(n). Lemma 2 establishes the base

, n = 3. So assume the assertion holds for some 3. Let G # Kn+3 be an (I# + 2)connected graph

with a I-factor. Let t be a totally covered vertex, and let tu be an edge.

Graph G - {?, u) is n-connected and has a l-factor. Further, it is not A&+, . For othenvise, each vertex w in C - (t, u) is adjacent to both t and u, since w has degree at least n + 2 in G. This implies G = K,r+3 9 con- trary to assumption.

The inductive assertion shows F(G - {t, u}) 2

; g(n). Since tl ‘. F(G) 2; .p(r

tk ic de ai least n + 2 vertices u, ?\ ,!CD

Theorem 5: Let n > 6 oe even. Th,en j (ra) 2 i 4

(n). Y;roof: We first show for n = 6, f(6) > i g(6) - 60.

Let G be a 6-connected graph wit11 a l-factor. It is easy to see a totally covered vertex r has at least sly edges tu such that G - {t, u) # P6. So Lemma 3 shows F(G) > 6 . BO = 60, whence f(6)! 2 60.

The general case now follows by induction, with n = 6 as the base. QED

Theorem 5 gives an exact bound for n = 6, as shown by the party graph f8. We conjecture that for any even n > 2, f(n) = F(P,+z).

References

[ 1 ] L.W. Beineke and M.D. Plummer, On the 1-factors of a non-separable graph, J. Comb. Theory 2 (1967) 285-289.

121 F. Harary, Graph Theory (Addison-Wesley, Reading, Mass., 1969).

[3] J. Zaks, On the l-factors of n-connected graphs, 9. Comb. Theory il(l971) 169-180.

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