hypercube subgraphs with minimal detours

Hypercube Subgraphs with Minimal Detours

Pal Erdos HUNGARIAN ACADEMY OF SCIENCES

Peter Hamburger* Raymond E. Pippert William D. Weakley

DEPARTMENT OF MATHEMATKAL SCIENCES

FORT WAYNE, INDIANA 46805 e-mail: hamburge@cvax. ipfw. indiana. edu

e-mail: pippert@cvax. ipfw. indiana. edu e-mail: weakle y@cvax. ipfw. indiana. edu

INDIANA UNIVERSITY-PURDUE UNIVERSITY FORT WAYNE

ABSTRACT

Define a minimal detour subgraph of the n-dimensional cube to be a spanning subgraph G of Qn having the property that for vertices 2, y of Qn, distances are related by dG(z, y) 5 dQ,(z,y) + 2. Let f(n) be the minimum number of edges of such a subgraph of Qn. After preliminary work on distances in subgraphs of product graphs, we show that

f(n)/lE(Qnl < 46. The subgraphs we construct to establish this bound have the property that the longest distances are the same as in Qn, and thus the diameter does not increase.

We establish a lower bound for f(n), show that vertices of high degree must be distributed throughout a minimal detour subgraph of Qn, and end with conjectures and questions. 0 1996 John Wiley &Sons, Inc.

1. INTRODUCTION

Although the hypercube has been studied for its own sake, it has attracted special interest in recent years for its properties as the underlying graph of a network, especially a

* Supported by a Purdue Research Foundation Summer Faculty Grant. Journal of Graph Theory Vol. 23, No. 2, 119-1 28 (1 996) 0 1996 John Wilev & Sons, Inc. CCC 0364-9024/96/020119-10

120 JOURNAL OF GRAPH THEORY

distributed-processing computer network. In this context, parameters involving connect- edness and distance are of special interest; it is well known that the n-dimensional cube Qn has connectivity n, edge-connectivity n, and diameter n. Another useful quantity is the domination number, which has been shown [8, 91 to be IV(QTL)l / (n + 1) when n = 2k - 1 for some positive integer k.

Of major interest, particularly in computer networks, is the effect of the removal of a set of vertices or edges, corresponding to inoperative microprocessors or broken connec- tions between them. Parameters which measure changes in other parameters induced by such breakdowns fall under the classification of leverage, introduced by Bagga, Beineke, Lipman, and Pippert [2], and this is the area of our present work. Previous results on leverage include establishing the edge-integrity of Qn, shown to be 2" (Qn is an honest graph) [l]; the integrity, shown surprisingly to be 0(2"logn/fi) [31; and the fact that the removal of fewer than n - 1 edges cannot increase the diameter, while the removal of n - 1 edges can increase it by at most one [2].

Clearly, the removal of any edge increases at least one distance (between the ends of the edge) by at least two, so the removal of k edges increases the distance sum by at least 2k. The maximum value of k for which this is achieved is k = 2n-1, obtained by deleting a perfect matching, no two edges of which lie in a common four-cycle [6]. The main question we address here is that of how many edges can be removed from Q T L without increasing the distance between any two vertices by more than two.

In our approach to this problem, we obtain a subgraph construction (Theorem 1) appli- cable to connected product graphs. For the subgraphs of Qn thus derived, we get results on the diameter and on how many distances have increased.

2. CONSTRUCTIONS AND UPPER BOUNDS

For any graph G and vertices v , w of G, let d ~ ( v , w ) denote the distance from u to w in G.

The following theorem on distances in subgraphs of product graphs is the basis of our constructions in hypercubes.

Theorem 1. Let H be a graph and let i be a positive integer. Let T C V ( H ) and U & E ( H ) be subsets chosen so that for every ordered pair (q , u2) of vertices of H , there is a walk W from v1 to v2 satisfying

(a) each edge of W is in U, (b) some vertex of W is in T , and (c) the length of W is at most d~ (v l , u2) + i.

Then for any connected graph L there is a spanning subgraph G of H x L such that

IE(G)I = IT1 . IE(L)I + IUI. IV(L)I

and for any vertices z,y of H x L,

Proof. Suppose that we have T, U satisfying the hypotheses. If we regard H x L as a copy of L in which each vertex has been replaced by a copy of H , the idea of

MINIMAL DETOUR SUBGRAPHS 121

FIGURE 1. For H = Q3 and i = 2, (a) shows sets T (circled vertices) and CJ (solid edges) satisfying the hypotheses of Theorem 1. Then with L = Q1 we have H x L z Q4; the induced subgraph G of Q4 given by Theorem 1 is shown in (b).

the construction is to let G be the subgraph induced by those edges in each copy of U together with those edges that join corresponding vertices of T in adjacent copies of H. More precisely, define a subset S of E ( H x L ) by S = {((w, wl), (w, w2)) : w E T, (w1, w2) E E ( L ) } U { ( ( q , w ) , ( w ~ , w ) ) : (w1,w2) E U,w E V ( L ) } . Let G be the subgraph of H x L induced by S; clearly G has the desired number of edges. That G is a spanning subgraph of H x L will follow when we establish (1) for all vertices z, y of H x L.

For some vertices ~1,212 of H and w1,w2 of L we have z = (w1,wl) and y = (2)2,w2).

Choose a walk W from v1 to w2 in H satisfying the hypotheses of the theorem, say w1 = zo 4 z1 -+ . . . + zd = 02, where d is the length of W. Let m be an index such that z , E T.

Choose a path of length dL(w1, w2) joining w1 to w2 in L, say w1 = TO + TI + . . . +

Tb = wz. We then construct a walk W' from z = (wl,wl) to y = (w2,w2) in G as follows:

( z , , T ~ ) = (z,,,w2) + (zm+1,w2) 4 . . . -+ (zd,w2) = ( ' u2 ,w~) . The length of W' is (W1,Wl) = (zo,w1) + (Zl,Wl) + ' . . + (zm,w1) = ( z m , ~ ~ ) + (zm,T1) -+ . . ' +

d + d ~ ( w l , w 2 ) , whichisatmost d ~ ( w 1 , ~ 2 ) + i t d r , ( w 1 , ~ 2 ) = d ~ ~ ~ ( z , y ) + i . I

In Figure 1 we show an example. Our particular interest here is in Theorem 1 with i = 2. (Note that if H and L are

bipartite graphs, then so is H x L, and we need consider only even values of i.) In this situation, for each vertex w of H there is a walk from w to w of length at most two and passing through a vertex t of T. This implies d ~ ( w , t ) 5 1, and thus T is a dominating set of H. Conversely, if T is a dominating set of H and U = E ( H ) , then T and U satisfy the hypotheses of Theorem 1 with i = 2.

For a graph D , let D" denote the n-fold product of D. It is easily proved by induction that IV(Dn)l = IV(D)I" and IE(D")J = nlV(D)I"-lJE(D)(. In the situation of Theorem 1, suppose H = D" and L = D"-". Define ratios T, = ITl/lV(H)I and T, = lUl/lE(H)l. A short calculation gives


TABLE I . For n I 7, we give the upper bound for f(n) from Proposition 2 and the lower bound from Theorem 12; these coincide for n 5 4.

n 1 2 3 4 5 6 7

f (n) 1 3 8 18 39:40 82:88 173:192

From now on, we take i = 2 in Theorem 1 and study the case D = K2, so DTL % Qn, the n-dimensional cube.

Definitions. A spanning subgraph G of Qn is a minimal detour subgraph of Q7, if for each pair of vertices x, y of Qn we have

d G ( x , 13) 5 dQrL (2, Y) + 2- ( 3 )

Let f ( n ) denote the minimum number of edges of such a subgraph of Qn. For a spanning subgraph G of Qn to be a local detour subgraph, we require only that

inequality (3) be satisfied whenever x and y are adjacent in Qn. Let f i(n) denote the minimum number of edges of any local detour subgraph of Q,,.

Clearly every minimal detour subgraph of QTL is also a local detour subgraph, and f i (n) 5 f ( n ) for each n.

Proposition 2. For each positive integer n,

f (n) / lE(Qn>l 5 (n + 5)/4n.

Proof. It is easy to see that f(1) = 1 and f ( 2 ) = 3, and thus that the bound holds for n = 1,2. For n 2 3, let G(n) be the subgraph of Qn given by Theorem 1 with H = Q3, L = QnP3, and the sets T , U shown in Figure l(a). Here r, = and T, = 5, so

I In Figure l(b) we show the graph G(4). For n 5 7, Table 1 gives the upper bound of

f(n) from Proposition 2. We now seek a better upper bound of f ( n ) for large n. From (2) and the discussion after

Theorem 1, it is clear that dominating sets T of Qm which give smaller values of the ratio T, will be useful. J. G. Mauldon [8] and S. K. Zaremba [9] showed independently that Qm has a perfect dominating set (every vertex covered exactly once) if and only if m = 2k - 1 for some positive integer k. These sets are the perfect single-error-correcting Hamming codes, discussed in 141, and they give r, = 2 T k . We give an inductive construction, without proof; it is convenient to identify V ( Q n ) with (Z2)" = (0 , l}n.

Theorem 3. Let TI = (0) and for k 2 1 set T k + l = {xyd : x , y E V(Q2k-1),d E {0,1>, x + y E T k and xd has even parity}. Then for each k, T k is a perfect dominating

Definition. For positive integers n, k such that n 2 2k - 1, let G(n, k) denote the minimal detour subgraph of Qn given by Theorem 1 with H = Q 2 k p 1 , L = Qn-2k+1,T = T k , and U = E ( H ) . Let G(n, 0) be Qn.

It is not difficult to see that G(n, k) has 2"-" vertices of degree n and its other vertices have degree 2k - 1.

(2) implies the desired bound.

Set Of Q2k-1.


Theorem 4. For each positive integer n,

Proof. For n = 1 the bound is trivial, so we will assume n > 1. For any non-negative integer k such that 2k - 1 5 n, since the graph G(n, k ) has 2n-k vertices of degree n and its other vertices have degree 2k - 1, we see G(n, k ) has fewer than (n2n-k + 2 n + k ) / 2 edges. Writing x = 2'", we have

JE(G(n, k))I < 2"-l((n/x) + x). If we regard the right side of this inequality as a function g of a continuous variable 2, by calculus we can show that the graph of g is everywhere concave up and the minimum of g occurs when x = fi. Now x must be rounded to an integral power of two such that 1 5 2 5 n + 1. Since 1 5 @ for n > 1 and 6 < n + 1 for all positive n, we may choose x to be within a factor of of fi. Then

which implies the desired conclusion. I We note that the bound of Theorem 4 is only superior to that of Proposition 2 for

n 2 62. In our construction of the graphs G(n, k ) , we used U = E ( Q 2 k - 1 ) in Theorem 1. We

do not know how much the bound of Theorem 4 can be improved by taking a smaller subset of E(Qp-,) to be U. Some light is shed on this by considering the problem of constructing local detour subgraphs of Q,, with a small number of edges.

The following notation is useful in the proofs of our next two results.

Notation. For each positive integer k and each vertex x of Q2~--1, let C ( x ) denote the unique vertex in Tk that is either adjacent or equal to x.

Theorem 5. For each positive integer n,

Proof. For each positive integer k let uk be a subset of E(Q2k-l) that contains all edges incident at vertices of Tk, and, for each pair t , a of vertices of Tk at distance 3, contains an additional pair of parallel edges from the copy of Q3 between t and z in Q 2 ~ - 1 .

For each integer n with n 2 2k - 1, let Gl(n, k ) be the subgraph of Qn obtained by the construction of Theorem 1 with H = Q2~-1, L = Q n - 2 k + l , T = Tk, and U = u k . (Then Gl(n, 2 ) Z G(n)). We now show that Gl(n, k) is a local detour subgraph of Qn.

Let 2, y be vertices of Qn at distance one. For some vertices w l , v2 of Q2k -l and w1, w2

ofQ,-2k+1,wehavez= (vl,wl)andy= (v2,w2).Sinced(z,y) = linQ,,eithervl =u2 or w1 = w2. We examine these two cases.

If v1 = 712, then in Q n - 2 k + l we have d(wl,w2) = 1, and the path (vl ,wl) + ( C ( v ~ ) , w l ) ---f ( C ( v l ) , w ~ ) = (C(v2),w2) + (v2,w2) lies in Gl(n,k) since by our choice of u k , the vertices in the copies of Tk have full degree n in Gl(n, k ) . Thus in Gl (n, k ) we have d(x, y) 5 3 as desired.


If w1 = w2 then in Q2k-l we have d(w1,v2) = 1. If vi E T k for i = 1 or 2, then since (wt, w,) has degree n in Gl(n, k ) we are done. Otherwise d(C(vl), C(v2)) = 3 in Q2kP1, so by our construction U, contains eight edges from the copy of Q3 between C(wl) and C(w2), the six incident at C(vl) or C(v2) and an additional parallel pair. This implies (see Figure l(a)) that there is a path of length 3 in UI, between w1 and v2, so d(z , y) = 3 in Gl(n, k ) . This completes the proof that Gl(n, 16) is a local detour subgraph of Q-,.

Since TI; is perfect, each vertex in TI, is at distance 3 from (2k;1)/3 other vertices of T k , and a calculation gives T, = (1 + 22-k)/3 for Uk. Using this with T, = 2-k in Eq. ( 2 ) and writing z for 2k, we obtain

f i(n)/lE(Qn)l I (3n - 1)/3nz + 5/3n.

An argument similar to that of Theorem 4 then establishes the desired bound. I

It is possible to improve on the construction of the graphs G ( n , k ) , for k 2 3, by choosing some of the parallel pairs of edges for Uk in such a way that other edges may be omitted from UI,.

Returning to minimal detour subgraphs, we next show that for the subgraphs G(n, k ) of Qn, the longest distances are the same as in Qn.

Lemma 6. If w, w are vertices of Q2kP1 and no minimal length path from w to w passes through a vertex of T k , then d(v, w) 5 2"l - 1.

Proof. Let d = d(v, w). The set of minimal length paths from w to w induce a copy of Qd inside Q2k -'. Let 211, . . . , ?Id be the vertices adjacent to w in this subgraph and let a, be the place in which w and wi differ. Then a l , . . . , a d are distinct.

Writing vo for v, we are assuming that for each i , 0 5 i I d, the vertex C(vt ) is not in the copy of Qd between and w. This implies wi # C(vi) for each i and, if v, and C(vi) differ in place b , , that bo, b l , . . . , b d are all distinct from al, . . . ,ad.

Finally, if b, = bj for some i , j , then d(C(vi), C(vj)) = d(vi, wj) 5 2, which since TI, is perfect implies C(v,) = C(vj), and then wi = wj, so i = j . Thus a l , . . . ,ad, bo,. . . , b d are 2d+l distinct elements of ( 1 , . . . , 2k-1}, implying 2d+l 5 2'-1 and then d 5 2"l-1. I

Proposition 7. For vertices z, y of Qn, if d ~ , , (x, y) 2 n - (2"' - 1) then dC;(n,k)(x, 3 ) =

Proof. There are vertices w, w of Q2kP1 and vertices t , u of Q7L--2k+l such that z = vt dQn (2, Y).

and y = WPL. Then (measuring distances in the appropriate cube)

d(w, w) + d ( t , u) = d(z , y) 2 n - (2k-1 - l),

so

d(v, w) - 2k-1 2 (n - 2k + 1) - d ( t , u ) 2 0,

where the last inequality follows from the fact that d(t ,u) is a distance in Qn-2k+1. Thus d(v, w) 2 2'"-', so by Lemma 6 there is a minimal length path from v to w in QZk--l that

I

This result is sharp; that is, there are vertices at distance n - 2'-l in Qn for which the

passes through a vertex of TI,, which implies the conclusion.

distance in G(n, k) is greater by two.

Corollary 8. If k 2 2, G(n, k ) has the same diameter as Qn


Proof. Since G(n, k) is a minimal detour subgraph of Q T L , we need only insure that for vertices z, y of Q,, if dQn(z,y) 2 n - 1 then dC(, ,k)(z , y) = d ~ * (z,y). For k _> 2 this

It is natural to ask how many distances increase when we pass from Qn to a minimal detour subgraph. We answer this question for the graphs G(n) from the proof of Proposition 2 and for G(n, 5) when k 5 3.

For n > 2, let r (n) denote the ratio of the number of distances in G(n) that have increased to the total number of nonzero distances in Qn. Let r (n ,k ) be the analogous ratio for G(n, k).

Proposition 9. For n > 2, r (n) = (3.2”-l - 5)/(2”+2 - 4) < 12/32.

follows from Proposition 7. I

For n > 1,r(n, 1) = (2,-’ - 1)/2(2” - I) < 8/32. For n > 3, r(n, 2) = 9(2n-3 - 1)/4(an - 1) < 9/32. For n > 7, ~ ( n , 3) = 161(2n-7 - 1)/8(2” - 1) < 5.04/32.

Proof. We will establish the values of r(n,lc) for k 5 3; the argument for r (n) is similar. Let n, k be positive integers with n > 2k - 1. For any vertices z, y of Qn, there are vertices v, w of Q2~-1 and vertices t, ‘ 1 ~ of Qn-2k+1 such that z = vt and y = wu. Then dG(n,k) (z, y) = d~~ (5 , y) + 2 if and only if t # u and there is no minimal length path from v to w in Q 2 k ~ l that passes through a vertex of Tk. (Otherwise dG(n,k)(z,y) = dQn(z, y)).

For the moment, say that an ordered pair (21, w) of vertices of Q 2 ~ - l is a bud pair if no minimal length path from 21 to w passes through a vertex of T k . It is clear that no vertex of Tk can be in a bad pair.

The vertex of Q1 that is not in TI is in one bad pair (with itself). This gives r(n, 1) =

Each of the six vertices of Q3 not in T2 is the first element of three bad pairs. Thus

It can be shown that each of the 112 vertices of Q 7 not in T3 is the first element of 23

(2%4)(2n-l - 1)/2”(2” - 1).

r(n, 2) = (2n-3)(6)(2n-3 - 1)(3)/2’L(2n - 1).

bad pairs, and therefore

+, 3) = (2n-7)(ii2)(2n-7 - i ) ( 2 3 ) / ~ ( 2 ~ - 1).

These values of r(n, k) reduce to those stated. I

It is interesting that for large n, the graph G(n,3) has about half as many edges as G(n, 2), but G(n, 3) has many fewer distances increase from those in Qn.

3. CONSTRAINTS AND LOWER BOUNDS

We now show that, roughly speaking, no vertex of a minimal detour subgraph of Qn can be too far from a vertex of high degree; this is reminiscent of the “hub” system used by many airlines and delivery companies. We then derive a lower bound for f i (n ) , and thus also for f(n).

we may associate to any edge (2, y) of Qn a direction rn, namely the index of the place in which vertices z and y differ. For any subgraph G of Qn and vertex z of G, let Sc(z) denote the set of directions of those edges of G incident at x. Let N G ( x ) denote the set of vertices adjacent to z in G.

Say that a path of length three in Qn is a three-side if its end edges have the same direction. (Such a path may be regarded as three sides of a square.)

After choosing an identification of V(Qn) with (0,


Proposition 10. Let G be a local detour subgraph of Qrl. For any vertex z of G,

u v t N n (X 1

SG(?4) = ( 1 , . . . n}.

Thus either z or one of its neighbors has degree at least fi. For any m in { 1,. . . , n}, there is a vertex z of Q7, that differs from z only in

place m. If (2, z ) is an edge of G, then z E N G ( z ) and m E SG(Z). If (2, z ) is not an edge of G, then since G is a local detour subgraph of QTL, there is a three-side P joining z and z in G. The middle edge of P has direction m and is incident with a neighbor of z. This establishes the set equation, and from that equation we have

Proof.

YE NG (XI

which implies the rest of the proposition. I

The last statement of Proposition 10 generalizes to minimal detour subgraphs.

Theorem 11. For each positive integer m, there is a positive constant c such that for every positive integer n 2 m, minimal detour subgraph G of Q n , and vertex w of G, there is a vertex w of G such that

d ~ , ( u , w) 5 m and deg,(w) 2 cnm/(m+l).

Proof. Proposition 10 implies that for m = 1 we can take c = 1. So we may assume n 2 m, 2 2, and let G be a minimal detour subgraph of Qn. Let v be a vertex of G and let k be the maximum degree in G of any vertex whose distance from v in Qn is at most m.

We look at paths from w to those vertices at distance m from w in Qn. There are (;L) such vertices; since G is a minimal detour subgraph, each of these can be reached from ZI by a path in G having length either m or m + 2. For these path lengths, we wish to bound the number of end vertices possible.

If a path from w to a vertex at distance m has length m, then each of the vertices in the path has degree at most k . Since two path edges are incident at each internal vertex of the path, there are at most k ( k - l)m-l end vertices reached by such paths from v.

If a path from v to a vertex at distance m has length m + 2, it includes exactly one step toward v. This step originates at a vertex of distance at most m + 1 from v, and should not retrace the previous step (else the path's final vertex is reached by a path already considered), so it can be made in at most m ways. Thus there are at most k(k - 1)-m additional end vertices reached by such paths. Thus

Since n 2 m, we have

so ml-m m n /m! 5 k(k - ~ ) ~ - - l + k ( k - ~ ) ~ m < /cm+'m,


and then [m-"/,!]ll(m+l)nml(m+l) k,

which completes the proof. I

Using the inequality m! < e[(m + l ) / e I m f 1 (see [7, page 17]), one can show that for any integer m > 1, the value of c produced in the above proof exceeds 2.45/m(m + 1).

Theorem 12. Let G be a local detour subgraph of Qn. Then

Prooj Since G is a local detour subgraph of Qn, for each edge e of Qn that is not in G we

may choose a three-side P(e) that joins the ends of e in G. Note that if e1,e2 are edges not in G, then P ( e l ) , P(e2) can share at most one edge (otherwise their union would be a square in G including el and e2). Thus a vertex of degree k in G can occur as an internal vertex of at most (!j) of the chosen three-sides. Then by counting chosen three-sides we have

For each k , 1 5 k 5 n, let bk denote the number of vertices of degree k in G.

k=l '-' k=l

which gives

n

k=l

We note one further constraint on G, namely b, 2 bl. This follows from the fact that if vertex 'u has degree one in G and its sole incident edge has direction m, then for each i # m, 'u is one end of a three-side in G whose middle edge has direction i; thus the sole neighbor w of w has degree n in G, and each neighbor of w other than 'u has degree at least two in G. Therefore, each vertex of degree one is adjacent to a vertex of degree n, which is adjacent to no other vertex of degree one.

We can now formulate a linear programming problem whose solution gives a lower bound for IE(G)I : minimize (C kbk) /2 subject to (41, b, 2 b l , bk = 2", and b k 2 0 for each k. By checking the vertices of the feasible region, the minimum for n > 3 can be shown to occur when bl = b, = 2"(2n - 6)/(n2 + n - l o ) , b2 = 2" - bl - b,, bk = 0 otherwise, and this gives the stated bound for IE(G)I. For n 5 3, similar checking gives the same bound. I

For small n this bound is good, as shown in Table I.

4. QUESTIONS AND CONJECTURES

In the minimal detour subgraphs G(n) of Q, constructed to prove Proposition 2 for n > 2, half the vertices have degree two and one-fourth have degree one. This seems a lot of low-degree vertices.


Conjecture. For n > 2, let G be a minimal detour subgraph of Qn with IE(G)I = f(n). Then at most three-fourths of the vertices of G have degree below three.

We believe that the upper bound for f (n ) given by Theorem 4 is better than the lower bound from Theorem 12.

Conjecture. The function f(n)/2" is unbounded.

Question. Is fi2"-' the order of magnitude of f(n)?

We have some evidence (Table I) that &(n) = f ( n ) for small n.

Question. Is f i = f ?

We finish with some speculations on diameter, occasioned by the work leading up to Corollary 8. Let S(n) be the collection of all spanning subgraphs of Qn having diameter n. Let A(G) denote the maximum vertex degree occurring in the graph G.

Question. Is min{A(G) : G E S(n ) } unbounded?

It is not difficult to construct a graph [5] in S(n) having size

2rL + ( ,;2, ) - 2.

Problem. Find good bounds for min(lE(G)\ : G E S(n)}.

References

K. S. Bagga, L. W. Beineke, M. J. Lipman, and R. E. Pippert, On the edge-integrity of graphs, Congress. Numel: 60 (1987), 141-144.

K. S. Bagga, L. W. Beineke, M. J. Lipman, and R. E. Pippert, The concept of leverage in network vulnerability, in: Graph theory, combinatorics, and applications (ed. Y. Alavi et al.). Conference Proceedings, Wiley, New York (1991), 2940.

L. W. Beineke, W. D. Goddard, P. Hamburger, D. J. Kleitman, M. J. Lipman, and R. E. Pippert, The integrity of the cube is small, J. Combin. Math. Combin. Comput. 9 (1991), 191-193.

E. R. Berlekamp, Algebraic coding theory, McGraw-Hill, New York (1968). N. Graham and F. Harary, Changing and unchanging the diameter of a hypercube, Discrete Appl. Math. 37/38 (1992), 265-274.

P. Hamburger, R. E. Pippert, and W. D. Weakley, On a leverage problem in the hypercube, Networks 22 (1992), 435439. P. P. Korovkin, Inequalities, Blaisdell Press, New York (1961).

J. G. Mauldon, Covering theorems for groups, Quart. J. Math. Oxford (21, 1 (1950), 284-287. S. K. Zaremba, A covering theorem €or abelian groups, J. London Math. Soc. 26 (1950), 71-72.

Received May 28, 1996

hypercube subgraphs with minimal detours

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