A shortness exponent for r-regular r-connected graphs

A Shortness Exponent for r- Regular r-Connected Graphs

Brad Jackson UNIVERSITY OF CA LlFORNlA

SANTA CRUZ, CA 95064

T. D. Parsons* PENNSYLVANIA STATE UNIVERSITY

UNIVERSITY PARK, PA 16802

ABSTRACT

Let r l 3 be an integer. It is shown that there exists & = &(r), 0 < & < 1, and an integer N = N(r) > 0 such that for all n 2 N (if r is even) or for all even n 2 N (if r is odd), there is an r-connected regular graph of valency r on exactly n vertices whose longest cycles have fewer than nE vertices.

1. INTRODUCTION

Our use of “graph” excludes loops and multiple edges. For any graph G, let n( G) be the number of vertices of G , and let c( G) be the length of any longest, cycle of G (if G is not acyclic). Then s(G) = c(G)/n(G) is the “shortness coefficient,” and a(G) = logc(G)/logn(G) is the “shortest exponent” of G. For any infinite family F of graphs, the shortness coefficient is s(F) = lim inf(s(G) 1 G E F} and the shortness exponent a(F) = lim inflcr(F) 1 G E 4.

Let r 2 3 be an integer, and let F, be the family of all regular r-valent r- connected graphs. G.H.J. Meredith [7] was the first to construct non- Hamiltonian graphs G E F, for every r 2 3, and his results implied s(F,) < 1. The problem of whether s(F,) = 0 for any r > 3 remained open; recently, we proved s(F,) = 0 for all r 2 3, but were unable to prove the stronger result that o(F,) < 1. (See [ 5 ] . )

Define l,(n) = min{c(G) 1 G CZ F, and n(G) = n). It follows from a theorem of Harary [4] that the domain of 1, is the set of all integers n 2 Y -t 1 if r is even or the set of all even integers n 2 r + 1 if r is odd. Bondy and

(see [31.)

*Research partially supported by NSF grant MCS-80-02263.

Journal of Graph Theory, Vol. 6 (1 982) 169-1 76 o 1982 by John Wiley B Sons, Inc. CCC 0364-9024/82/020169-07$01.70

170 JOURNAL OF GRAPH THEORY

Simonovits [2] have shown that exp(a log”2n) < Z3(n) < nb for some Q > 0, 0 < b < 1, and all sufficiently large (even) integers n. They conjectured nc < Z3(n) for some constant c > 0, and this problem remains unsolved. Laszlo Babai [ 11 has asked whether for any r > 3 we have Zdn) < nE for some constant E , 0 < E < 1 , and all sufficiently large n in the domain of I,. In our earlier paper [5], we proved lim,,,[ZXn)ln] = 0 (where n -* m through even n if r is odd) for all r 2 3. We shall now improve that result to show ZXn) < n E (for large n ) , which answers Babai’s question affirmatively, and implies that a(F,) < 1 , for each r l 3 .

2. A RECURSIVE CONSTRUCTION

Let r 2 3 be an integer which will remain fixed throughout the following discussion. We shall construct recursively a sequence (Gi ) of non- Hamiltonian graphs Gi E F, with lim inf,,, a(Gi) < 1. The construction is similar to one in our paper [ 5 ] . Later, we will use other constructions based on (Gi) to show Z,(n) < nE, for all large n.

Let Go be any non-Hamiltonian graph in F, such that n( Go) is even. Such graphs always exist; indeed, Meredith [7] constructed such graphs having lO(2r - 1) vertices. Let no = n(Go) and co = c(G0); then co < no. If Gi has already been defined, then we choose v i to be some vertex of Gi which lies m a longest cycle of Gi, we let wi Wi, . . . , wf be some fixed ordering of the neighbors of ui in Gi, and we let Hi be the g~ aph obtained from r - 1 disjoint copies ( Gi - v i ) l,. . . , ( Gi - v ~ ) , - ~ of Gi - vi and r new vertices u l,. . . , u, by joining vertex uj to the copy of vertex wj in each (Gi - v i ) k for k = 1,. . . , r - 1 and for j = 1,. . . , r. (See Fig. 1.) The graph Hi thus has r -I- ( r - l)[n(Gi) - I] vertices.

We now construct the graph Gi+l by a certain “substitution” of a copy of Hi at each vertex v of Go. Namely, if v is a vertex of Go ana el, e2,. . . , e, is some fixed ordering of the edges of Go which are incident to v , then we remove u from Go so as to leave r “pendant” edges (minus the endpoint v ) , and we reconnect these pendants el, e2,. . . , e, to the vertices u l , u2 , . . . , u, (respectively) of a copy H,(u) of H,. (See Fig. 2.) We carry out such a substitution of a copy H,(v) of Hi for vertex u E V(Go) at every vertex v of Go, and the resulting graph is Gi+l. Note that Gi+l depends upon the various orderings el, e2, . . ,. , e, of edges incident to vertices v of Go, and it depenas on the ordering Wi, wi, . . . , wi of the neighbors of vi in G j ; we let (GiJ be any par- ticular such sequence, which hereafter remains fixed. We refer to the disjoint induced subgraphs Hi(v) , for v E V(Go), as “supervertices” of Gi+l ; if each such supervertex of Gi+l is shrunk to a point, then we obtain a copy of Go.

It is not difficult to show that Gi E F, and Gi is non-Hamiltonian: we shall omit the proofs, which follow from our results [6] involving general constructions for non-Hamiltonian graphs in F,.

A SHORTNESS EXPONENT 171

FIGURE 1. T h e graph Hi.

Let ni = n(Gi) and ci = c(Gi). Then ni+l = n,Jr + ( r - l)(ni - l)] = no[ 1 + ( r - l )ni] . Also, it is easy to see that ci+l 2 coci > 2ci; indeed, Hi contains a path of length ci from some vertex up through copy (Gi - v i ) ~ to some vertex u q , and using some cycle of length c in Go, we can visit the supervertices Hr(v) corresponding to successive vertices on the cycle in Go and then travel within each H,{v) so as to include a path of length 2 ci in that Hi(u), giving a cycle in Gi+, of length at least coci. It follows that the sequence (ci} has ci -+ 00 monotonically as i + 00. We now investigate more closely the behavior of longest cycles in Gi+'.

Let C be a longest cycle in Gi+, , and let C* be the closed trail in Go which

vertex v in Go

FIGURE 2. ( a ) Vertex v in Go; (b) H;(v) in Gi+,.


arises from C by shrinking all supervertices H,{v) of Gi+l to single vertices v of Go. Let U be the set of vertices v of Go which lie on C*, that is, those vertices v of Go for which C n HXv) # q5. For v E U, iet c(v) be the number of vertices of C in H,{v), and consider the paths induced by C inH,(v). Each vertex uj of H,{u) (see Figs. 1,2) which lies on C satisfies exactly one of the following two conditions:

(a) uj is incident to an edge of C which is not in H,(v), and to an edge of C

(b) uj is incident to two edges of C which are both in HXv). in HI{ v);

Let a(v) and b(v) be the numbers of vertices u l , u2,. . . , u, of H,(v) which satisfy conditions (a) and (b). Then, obviously, for v € U we have

(i) a(v) is even and a(v) > 0. (ii) a(v) + b(v) 5 r. (iii) The number of edges of C joining u l , u2, . . . , u, to the other vertices of

HXv) is a(v) + 2b(v).

Now in H,{v) identify the vertices u l , u 2 , . . . , u, into a single vertex u, to form a new graph Hy(v), in which those edges of C n H,(v) induce a closed trail T such that every vertex of T, except possibly u, must have valency 2 in T. ( See Fig. 3, in which edges of T are represented by solid lines and curves.)

Now T is composed of some cycles with a single common vertex u, and by (iii) there are at most $a(v) + b(v) such cycles. Also, each component of T - u is a path in some (Gi - u;)~, and since this path plus v i give a cycle in Gi , each such path has at most ci - 1 vertices. Therefore C contains at most [&v) + b(t,)](c, - 1) vertices of the (Gi - Y ; ) ~ , k = 1,2, . . . , r - 1 in H,{v), and so C contains c(v) F a(v) -I- b(v) + [iu(v) + b(v)](ci - 1) vertices of H,{v). Now by (i) and (ii) we have a(v) 4- b(v) 5 r and ;a(.) + b(v) $2) + ( r - 2) = r - 1, so that c(v) 5 r 4- ( r - l)(ci - 1). Furthermore, if a(v) > 2, then a(v) 2 4 so that in this case h ( v ) -I- b(v) I x4) + ( r - 4) = r - 2 and C(U) I r + (7 - 2)(ci - 1).

Consider the closed trail C* in Go, which was defined previously. If C* is a cycle, then 1 U ( 5 co, so that

ci+ 1= I c I = z c ( v ) s co[r + ( r - I)(c, - l)]. v ELI

On the other hand, if C* is not a cycle, then some a(v) > 2 for t, E U, and since I U ( 2 1 V(Go)I = no, in this case we get

ci+, = 1 c / = z c ( u ) = z c ( U ) + C ( v ) l ( n o - l ) [ r + ( r - l ) ( C i - - l ) ] u e u u f w


+ c ( v ) I (no - l)[r + ( r - l)(Ci - l ) ] + [ r + ( r - 2 ) ( c ; - l ) ]

= no[r + ( r - l ) ( C i - 1 ) j - ( C i - 1 ) .

NOW no - c0 I 1 , SO

( n o [ r + ( r - l)(ci- l ) ] - ( c i - 1)) co[r+(r- l ) (c j - l ) ]

2 r + ( r - l)(c; - 1) - ( c ; - 1) 1 r + ( r - 2) (c i - 1) > 0.

Therefore in either case (P a cycle, or not) we find that

where

c i - 1 - c ; - 1 a , = + - ( r - 1)c; ci no(r - l ) c j

Now if i + 00, then ci -* OJ monotonically, so that Ji -, 1 - l/no(r - 1). L e t j be some fixed positive integer such that i z j implies

1 1 6,s 1 + = a; no(r -. 1 ) 2no(n - 1 )

then 0 < 8 < 1, and for i 2 j we have c ; + ~ I [ndr - 1)6]ci. By induction, cj+k 5 [ndr - l)Slkcj for all k 2 0, so logcj+k I k logndr - 1)s + logj. Now ni+l = no[l + ( r - l)ni] > no(r - l)ni, SO that nj+k 2 [ndr - l)lkni for all k 2 0 and 10gn~+~ 2 k logno(r - 1) + lopj. Therefore, for k 2 0,

l0gcj+k < k l o g n o ( r - 1)s + l0gcj - -

lowj+k k logno(r - 1 ) + l o p j ’

and the latter quotient has limit logno(r - 1)8/logndr - 1 ) < 1 as k -, OJ.

Therefore there exists u, 0 < u < 1 , such that for all sufficiently large k, logc,:+k/lognj+k < (T. This implies that ci < np for all sufficiently large i, and lim mfi,, a(Gj) 2 u < 1. This proves our first Theorem.

Theorem 1. For all r 1 3, we have a(l;,)< 1 .


FIGURE 3. H)(v) with trail T.

3. THE BEHAVIOR OF lr (n).

Recall that Zin) is the minimum of the longest cycle lengths c(G) over all G E F , having exactly n vertices [and ZXn) is undefined only if n < r + 1 or both n and r are odd]. We now prove

Theorem 2. For every r 2 3 there exist E = ~ ( r ) such that 0 < E < 1, and an integer N = N(r) > 0, such that 1Xn) < n' for all integers n 2 N when r is even or for all even integers n 2 N when r is odd.

Boo$ Let Gi be the sequence of graphs constructed in the previous section. Since no is even and ni+, = no[l + ( r - l )n i ] , it follows that ni is even for all i 2 0. We have shown that there exists a constant 8, 0 < 8 < 1, such that for all sufficiently large i, we have ci < n:-'. Let di = ( r - 1)n; + r + 1. Choose an integer Zo > 0 such that all the following conditions hold for every i 2 I,:

(iv) ci/ni= si < n,', (v) [log(4r6ni) 4- (2 - b)logni]/[log(r - 2)2 4- 2 lognil < 1 -as, (vi) di+l(di+l - 2) + 2r + 1 < r2n;+1, (vii) ( r - 2)2n; < di(di - 2) + 2r + I.

Let us verify that such an integer Zo exists. Since ci < ntV6 holds for all large i. conditions (iv) is satisfiable. Now ni -* 00 as i -, 00, and this easily


gives (v), (vi), and (vii) for all sufficiently large i. Thus such an integer I. exists.

Using the same techniques as in Section 4 of our paper [5], we see that it is possible for all i greater than or equal to some integer Il to use the graph Gi to construct graphs GI E F, with shortness coefficient s(G;) < si4rZ and with exactly n vertices, for all integers n > di (d i - 2) + 2r + 1 when r is even or for all even integers n > di(di - 2) + 2r + 1 when r is odd.

Let I = max(Io, 11), and consider all those graphs GI for i 2 I and for n such that dt(di - 2) + 2r + 1 < n zdi+l(di+l - 2) 4- 2r + 1. Then c(Gf)/n = s(G7) 5 si4r2 < n,'4r2, so that c(GI) < n;'4?n. Therefore,

8 4

< 1 --.

Here we have used conditions (iv)--(vii), plus the easily proved fact that nifl =< nomi. Letting E = 1 - 48 andN= dI(dI - 2) + 2r + 2, we have now proved Theorem 2.

References

[ l ] L. Babai, problem posed at The Research Workshop in Algebraic

121 J.A. Bondy and M. Simonovits, Longest cycles in 3-connected 3-regular

[3] B. Griinbaum and H. Walther, Shortness exponents of families of

[4] F. Harary, The maximum connectivity of a graph. Proc. Natl. Acad. Sci.

Combinatoncs, Simon Fraser University, 1979.

graphs. Canad. J . Math. 32 (1980) 987-992.

graphs. J Combinatorial Theory Ser. A 14 (1973) 364-385.

U.S.A. 48 (1962) 1142-46.


[5] B. Jackson and T.D. Parsons, Longest cycles in r-regular r-connected graphs. J. Combinatorial Theory. To appear.

[6) B. Jackson and T.D. Parsons, On r-regular r-connected non- Hamiltonian graphs. Bull. Australian Math. Soc. 24 (1 98 1) 205-220.

171 G. H. J. Meredith, Regular n-valent n-connected non-Hamiltonian non- n-edge-colorable graphs. J. Combinatorial Theory Ser. B 14 (1973) 5 5-60.

A shortness exponent for r-regular r-connected graphs

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