ON A FAMILY OF PLANAR BICRITICAL GRAPHS

Transcription

1 ON A FAMILY OF PLANAR BICRITICAL GRAPHS By L. LOVASZf and M. D. PLUMMER [Received 19 August 1973] 1. Introduction A l-factor of a graph G is a set of independent lines in 0 which span V(O). Tutte ([7]) gives necessary and sufficient conditions for a graph to contain a l-factor. A natural question to then ask is 'how many different 1-factors may a graph possess?' Kotzig ([3]) and, independently, Beineke and Plummer ([1]) proved that any 2-connected graph with a l-factor contains at least two of them. Zaks ([10]) generalized this result by proving that any ^-connected graph (k ^ 2) has at least &(&-2)(ifc-4)... 1-factors and this is in a sense, best possible, since the complete graphs K k+1 {k odd) contain exactly k(k 2)(k 4)... 1-factors. On the other hand, Lovasz ([4]) has shown that many ^-connected graphs have at least k\ 1-factors. More precisely, any such ^-connected graph with a l-factor and which is not bicritical has at least k\ 1-factors. A graph is bicritical if for every pair of points u and v in G, G\{u,v} has a l-factor. In view of the Lovasz result above, it becomes natural to ask 'what is the lower bound on the number of 1-factors a bicritical graph must have V In another paper, [5], the authors obtain a number of structural results on bicritical graphs in general which do provide a lower bound which is not thought to be best possible at all. In these studies, however, the authors were led, in particular, to the study of a subclass 34? of all bicritical graphs which seems interesting from several graph-theoretical points of view and for which a best possible lower bound has been obtained. We call # the class of Halin graphs. A graph H is a Halin graph if it can be constructed as follows: let T be any tree in which each non-endpoint has minimum degree 3. Embed T in the plane and construct a cycle through all the endpoints of I 7 in such a way that H = TuC remains plane (cf. Fig. 1). Why are such graphs of interest? Halin ([2]) uses them in his work on minimally 3-connected graphs (hence our name for 2/iP). Moreover, the cubic Halin graphs are in one-to-one correspondence (via duality) with the triangulations of the disc. Bondy and Lovasz (in unpublished work) proved that these graphs are almost pancyclic. They contain cycles of t Work supported in part by NSF Contract GP Proc. London Math. Soc. (3) 30 (1975)

2 ON A FAMILY OF PLANAR BICRITICAL GRAPHS 161 each length between 3 and j V{G) j except, possibly, one and this one exception must be of even length. FIG. 1 The subclass of 34? in which the graphs are cubic has been studied by Rademacher ([6]) and others with the purpose of enumerating them. Rademacher calls this subclass the based polyhedra and Polya calls them roofless polyhedra. Indeed, interest in these graphs may be traced all the way back to Kirkman. For our purposes, however, they are interesting for different reasons. Let cp{g) denote the number of different 1-factors contained in the graph 0. Since we seek a lower bound for <p(o) when 0 is bicritical, it is natural to study graphs which are bicritical and minimal with respect to this property; that is, 0 is bicritical, but for each line x, O\xia not bicritical. We shall show below that every even Halin graph is such a minimal bicritical graph. Moreover, we obtain as our main result (Theorem 4.2) a best possible lower bound for <p{h) when H is a Halin graph. In particular, if H is a Halin graph with p points and p is even, then <p(h) ^ f(jp 1), with the single exception of the graph of Fig. 18. This bound is shown to be best possible. In the final section we shall point out another possible reason for the study of the class «5f. 2. The even members of Jf are minimal bicritical To prove this statement, we shall make use of an unpublished result of Bondy. THEOREM 2.1 (Bondy). All Halin graphs H are 1-hamiltonian; thai is, both H and H\v have hamiltonian cycles for all points v. THEOREM 2.2. Every even Halin graph H is minivial bicritical. Proof. Let u,v be arbitrary points of H. Thus H\u has a hamiltonian cycle by Theorem 2.1 and hence H\{u, v} has a hamiltonian path which, since H is even, contains a 1-factor. Thus H is bicritical L

3 162 L. LOVASZ AND M. D. PLUMMER Let e = uv be any line in H. Then 0\e has a pair {e^ e 2 } of lines which separate it and hence, it cannot be bicritical (see [5]). 3. A useful subclass of 34? Let H = Tu C be a member of 2/?. LEMMA 3.1. Let V(H) = AuB, A ^ 0, B # 0, be a partition of V(H) such that the subgraphs induced by A and B (denoted by {A} and (B), respectively) are both 2-line connected. Then this partition is induced by a line of the tree T. Proof. Let C be the bounding cycle of H. First, we claim that A u B partitions G into precisely two paths. Suppose not. Then there exist points a v a 2 e A, b lf b 2 e B which separate each other on C. Since (A) and <JB> are connected, there is a path P A in (A} joining a x and a 2 inside C and a similar path P B in (B) joining 6 X and b 2 inside C. But then by planarity, P A np B^ 0, a, contradiction. This proves our lemma. Thus without loss of generality, we may assume that C is composed of path AnC with endpoints a x and a 2 together with BnC with endpoints b x and 6 2 (cf. Fig. 2), together with lines a^ and a 2 b 2. Now let P be the unique path in T joining b x and 6 2. We now claim that V(P) B. Suppose not; then there exists c e V(P)nA. Consider H\A. Then there are no two line-disjoint paths in <2?> joining 6 X and b 2, which contradicts the hypothesis that (B) is 2-line-connected and verifies the claim. Similarly, if Q is the unique path in T joining a x and a 2, V{Q) S A.

4 ON A FAMILY OF PLANAR BICRITICAL GRAPHS 163 Now let R be the unique path in T joining P and Q. Thus R is a single line by definition of #, and R determines a cut of H (together with a x b x and a 2 b 2 ). Moreover, by planarity and connectivity, all points of H on (Cn(A})uQ and interior to this cycle are in A. Similarly, those points of H on (Cn <» up and interior to this cycle are in B; this proves the lemma. DEFINITION. A bicritical graph G is contraction-reducible if there is a partition V(G) = AuB with A, \ B ^ 2 such that if two new graphs G A and G B are formed by respectively contracting B to a point and A to a point, then G A and G B are bicritical. Otherwise, G is contractionirreducible (cf. Fig. 3). A I B G' (a) G is contraction-reducible (b) G'is contraction-irreducible FIG. 3 Let H = T u C be any even Halin graph and x any line of T. Then T\x consists of two trees T x and T 2. If V{T X ) \ and V(T 2 ) are both even, we say that x induces an even cut of T. Otherwise x induces an odd cut. DEFINITION. An even Halin graph TI TuC has the even-cut property if each non-endline of T induces an even cut of T. THEOREM 3.2. The following are equivalent for any even Halin graph H = TuC:

5 164 L. LOVASZ AND M. D. PLUMMER (i) each inner point of T is joined to an odd number of endpoints of T, (ii) H has the even-cut property; (iii) H is contraction-irreducible. Proof, (iii) => (ii). Suppose that H has an odd cut V(H) = AuB induced by line e of T. Then G A and G B are both in 3f and hence bicritical, contrary to (iii). (ii) => (i). Suppose that (ii) is true and let u be any inner point of T. Let x = uv be any line incident with u. Then x separates T into two subtrees T u containing u and T v containing v and both subtrees have an even number of points. By parity, T u \u has an odd number of odd components and moreover each such odd component T' must be a single line. For suppose that T' contains more than one line; then if uw is the line of T joining u to a point w in T f, uw is not an endline in T and uw induces an odd cut in T, contrary to (ii). (ii) => (iii). Suppose that H is contraction-reducible, that is, V(H) = AuB, \A\,\B\>2 (of course both are odd), and H A and H B are both bicritical. Then the graphs (Ay and (B) each have the property that if any point is removed, the resulting graph contains a 1-factor. Clearly, then, each of (A} and <i?> is 2-line-connected and so, by Lemma 3.1, this partition is induced by a line of the tree T, contrary to (ii). (i) => (ii). Let x be any non-endline of T and consider the cut induced by x. This cut induces a partition V(H) = AuB. By (i) each inner point of T which lies in A is joined to an odd number of endpoints of T (in A) and hence A is even. Similarly B is even and the theorem is proved. We shall denote by 3P the subclass of all even Halin graphs satisfying any of the three equivalent statements of the preceding theorem. Our next objective is to determine a sharp lower bound for <p(h) when He 3T. DEFINITION. A subgraph F of a Halin graph H (= TuC) is an n-fan if F is a K(l, n), the centre of this star is an inner point of T incident with edges of F and one other edge, and the n other points of F appear consecutively on the cycle G. LEMMA 3.3. Let T be a tree with an even number (four, at least) of points and having at most one point of degree 2. Let P be a path through all the endpoints of T and suppose that G = TuP. Then <p(g) ^ 3 unless G is one of the graphs in the family & shown in Fig. 4.

6 ON A FAMILY OF PLANAR BICRITICAL GRAPHS 165 Fio. 4 Proof. The requirement that p ^ 4 is necessary; for if T = K 2, form G = TuP, where P is any path containing an odd number of lines and joining the endpoints of T. G is then an even cycle. Then G has exactly two 1-factors and G does not belong to *&. Let G o be a smallest graph such that the theorem is false. We may assume that T is not a K{\, n) for any n. Let F he a, fan in G o. Case 1. \E(F)\ ^ 3. Contract I 1 to a point and thus obtain a new smaller graph O' Q. Now O' 0 has a hamiltonian cycle C by Theorem 2.1. (a) E(F) is odd. Denote the contracted fan in G' o by /. Start with either line of C at / and take every other line of C until one returns (by parity) to the other line of C at /. One may extend this set of lines to a 1-factor F x of G o. On the other hand, start with a line of C not incident with / and take every second line of C as long as possible before returning to /. This set of lines extends to at least two different 1-factors of G o, say F z and F 3> and each is in turn different from F v (b) E(F) is even. Let u be the point of F not on P. Let a; be a line of G not incident with u, but incident with the fan F (cf. Fig. 5) and take every second line of C. By parity, no chosen line of this set S is incident with/and hence extends to at least two different 1-factors of G o. FIG. 5 Now consider E(C)\S. This set of lines extends to a 1-factor of G o in two possible ways, depending upon whether one or two lines of E(G)\S lies on P. However, in either case, we get at least three different 1-factors for G o. Case 2. Assume that all?i-fans of G o are 1-fans or 2-fans. Since T is not a K(l,n), we may assume that there is at least one 2-fan F. Contract F to get a smaller graph G' o. If G f 0 has at least three different 1-factors, each extends to a different 1-factor of G, and we have done. On the other hand,

7 166 L. LOVASZ AND M. D. PLUMMER if G' o has no more than two 1-factors, then since G' o is smaller than G o, by the minimality of G o, G' o must be one of the exceptional graphs of class &. Say G' o has the appearance of Fig. 6. FIG. 6 We claim that either a or b corresponds to the contracted 2-fan F. Otherwise, in G o, let F a be the end 2-fan containing a. Contract F a to the point a; then the resulting graph G' o is as before in class &. This is a contradiction, for G' Q contains a triangle (that is, the 2-fan F with a line of P joining its endpoints) and all points of this triangle have degree at least 3. Since no member of 3? has this property, our claim is proved. Thus the contracted fan F does correspond to either a or b and in either case G o e &, a contradiction; and the lemma is proved. We are now prepared for the main result of this section. THEOREM 3.4. If H e 2$ and \ V{H)\ = p, then <p{h) >p-l bound is best possible. and this Proof. (The proof is by induction on p.) Let L be a non-endline of the tree T in H = Tu G. Thus L induces a partition V{H) = A ub, where, by the even-cut property, (Ay and (By are both even. Suppose that (By has r points and hence that (Ay has p r points. Assume first that each of (Ay and (By is larger than a 3-fan. Then form two new graphs G 1 and G 2 by replacing (Ay and (By respectively with 3-fans (cf. Fig. 7). For i = 1, 2, and 3, let y^ denote the number of 1-factors of G x not containing Hue 6^ but containing both bfj, j ^ i. Similarly, let 8^ denote the number of 1-factors of G z not containing a i d i but containing both a^dj, j ^ i. Also, let y 0 (respectively 8 0 ) denote the number of 1-factors in (By (respectively <^4». Now each of G x and G 2 is a smaller member of 3& and hence by the inductive hypothesis, and, similarly, r + 3 = V(Q x )\-l <p{g 2 ) = S ^ p-r + Z.

8 ON A FAMILY OF PLANAR BICRITICAL GRAPHS 167 Next note that <J5> and <.4> are not members of the exceptional class ^ of Lemma 3.3, otherwise H would not have the even-cut property. Thus FIG. 7 both y 0 and S o are at least 3 by Lemma 3. We also claim that y x ^ 1. For consider #i\{& 2 >&3}- Since G x is bicritical, #i\{& 2 >&3} nas a 1-factor F x which by parity cannot contain b x c x. Hence {F x n (< > \ {6 2, 6 3 })) u {6 2 c 2,6 3 c 3, qcj is counted by y x. Similarly, y 2, y 3, S l5 8 2, and S 3 are all at least 1. Thus <p(h) = -7 =p-l, as desired. So we may assume that every inner line L of the tree T induces a cut such that one piece is a 3-fan. There are, then, only two possibilities. Case 1. Tree T has a bicentre (that is, two central points which are of course joined by a line). Let the bicentre line be ab. Then T consists of ab, a 3-fan at b, say, and a (2k + l)-fan at a for some positive integer k. Hence H has 2k+ 6 points (cf. Fig. 8). There are k +1 1-factors of H\{6, c v c 2, c 3 } each of which extends to two 1-factors of H. In addition, there are k 1-factors of H which contain 6c 2 and two others which contain ab. Thus <p(h) = 2(fc+l) + /c-l-2 = 3& + 4. Hence if k = 1, H has exactly p-l 1-factors, whereas if k > 1, <p{h) ^ p.

9 168 L. LOVASZ AND M. D. PLUMMEE Case 2. Tree T has only one central point v. We may assume there is a line x incident with v and with a non-endpoint of T, or else JET is a wheel and H- FIG. 8 we have done. Moreover, there must be two such lines (call the other y) or else T has a bicentre and we have done, by Case 1. In addition, if y = vw, w is the apex of a 3-fan or we have done by the first part of this proof. A similar argument applies when x = vu (cf. Fig. 9). FIG. 9 (a) If degv = 3, H is the 10-point graph of Fig. 10 which has ten 1-factors, so we have done. FIG. 10 (b) Hence assume degv ^ 4. In this case, replace any 3-fan by a single line L to obtain another member H o of 3$, where H o has p 4 points

10 ON A FAMILY OF PLANAR BICRITICAL GRAPHS 169 (cf. Fig. 11). Let the line of H incident with v that has been so removed be labelled L'. H-. FIG. 11 Let A o be the number of 1-factors of H o which contain line L, and X 1 be the number of those which do not. Now by the inductive hypothesis, A o + A x $5 (p 4) 1. Let A 2 be the number of 1-factors of H containing U. Thus <p(h) = A 0 + 2A 1 + A 2. Now form H x from H by removing the endpoints of L'. Clearly, H x contains at least three 1-factors. Moreover, by Lemma 3.3, A x ^ 3. Thus This completes the proof of the theorem. Moreover, we have also shown that the only members of jft with exactly # 1 1-factors are the wheels and the graph shown in Fig. 8 with k = The main result Of course, any line L of an arbitrary even Halin graph H induces a cut of H which may be odd or even. Suppose that L induces the odd cut A ujj. If neither <J.> nor (B} consists of a single point, we call the odd cut non-trivial, and if A \ ^ 5 and \B\ ^ 5, we call the odd cut proper. If L induces an odd cut equal to «-4>, <-B», the two subgraphs <.4> and <JS> are joined by exactly three lines: L = E 2, E lt and E 3, say. Of course, by parity in the case of an odd cut, any 1-factor of H will contain exactly one of the E i or all three. Consider the following construction. Let H be an even Halin graph and suppose that line y induces a proper cut of H; that is, a partition of V(H) = A u B is induced, where \A\ = r is odd, as is B \ = p r. Now form two new graphs H A and H B by replacing B and A respectively with triangles (cf. Fig. 12). Let y 0 be the number of 1-factors of H A containing a^, a 2 c 2, and a 3 c 3, y x be the number containing a^, but not a 2 c 2 or a 3 c 3, y 2 be the number

11 H A FIG L. LOVASZ AND M. D. PLUMMER containing a 2 c 2, but not a 1 c 1 or a 3 c 3, and y 3 be the number containing a 3 c 3, but not a x c x or a 2 c 2. Define 5 0, 8 V 8 2, and S 3 analogously with respect to H B and lines b x d x, b 2 d 2, and 6 3 d 3. We shall need the following lemma. LEMMA 4.1. Let H be an even Halin graph which has at least one proper odd cut. Suppose, moreover, that for every proper odd cut of H, S?-o (y< ~ 1)(8< 1) = 0. Then H is one of the two graphs shown in Fig. 13. (a) FIG. 13 (b) Proof. Since H has a proper odd cut, suppose that H is as in Fig. 12 and the cut induced by y is as shown there. Then neither <J.> nor <.B> is a triangle. Form H A and H B as in Fig. 12 with the same labelling used

12 ON A FAMILY OF PLANAR BICRITICAL GRAPHS 171 there. It is easy to verify that since H is bicritical, so are H A and H B. It follows from this that y^ 1 and 8^ 1 {i = 0,1,2, 3). Without loss of generality we may suppose that y 1 = 1. Then <^4>\{a} has a unique 1-factor F v Thus (A}\{a} is not 2-connected ([1], [3]). Hence a x and a 2 are adjacent and dega 2 = 3. Now if y 3 = 1, then a 2 and a 3 are adjacent and (Ay is a triangle, which contradicts the assumption that the cut was proper. Hence y 3 ^ 2. Hence S 3 = 1 since all four terms of the sum of the hypothesis are zero. For the same reason, one of y 2 or 8 2 equals 1, and by the symmetry of the y i and 8 it we may suppose that y 2 = 1. Let the points of <^4>\{a} which are adjacent to a x be a 2 and a' v and the point of <^4>\{a} adjacent to a 2 be a 2 (cf. Fig. 14). FIG. 14 Consider the (odd) cut induced by a 2 a 2. Suppose that this cut is also proper. Let y'^h^ (i = 0,1, 2, 3) be defined with respect to this new cut analogously to the y i} 8 i of the original odd cut. Since c v c 2, and c 3 span a triangle, y 3 ^ y 0. Also 83 ^ S 3 + S o > 1; so y 3 = 1, by the hypotheses of this lemma. Hence a B is adjacent to a' 2 and dega 2 = 3 as above. Also, recall that since y 2 = 1 by assumption, it follows that y[ = 1. So, as before, a[ is adjacent to a 2, which contradicts the assumption that the second cut was proper. Thus H A has the appearance of Fig. 15. Now consider H B. Recall that y 3 ^ 2 and hence that 8 3 = 1. Let F 3 be the unique 1-factor counted by S 3. As before, 6 3 is adjacent to b 2 and deg& 2 = 3, say 6 2 is adjacent to b' 2 (cf. Fig. 16). Moreover, we may assume that b 2 is not on the bounding cycle C of H = T u C since the original cut was proper. Thus H has the structure of Fig. 17. Now consider the cut induced by & If it is not proper, H is the graph of Fig. 13(a) and we have done. So this cut is proper. Again let y" i} S be defined with respect to this cut as were y t and 8 t with respect to the original

13 172 L. LOVASZ AND M. D. PLUMMER cut. One easily verifies that y[ = 2 = y r 2. Hence 8[ = 1 = 8' 2. Thus 0 must be the graph of Fig. 13(b); and the lemma is proved. FIG. 15 FIG. 16 H-- FIG. 17 We are now prepared to prove our main theorem. THEOREM 4.2. If H is a Halin graph with p points and p is even, then <p(h) ^ (p 1) unless H is the graph of Fig. 18. Moreover, this bound is best possible.

14 ON A FAMILY OF PLANAR BICRITICAL GRAPHS 173. / FIG. 18 Proof. The proof is by induction on p. If H e 2ft?, we have done, by Theorem 3.4. So suppose that H has an odd cut induced by a line y of the tree T in H = TuC. We adopt the labelling of Fig. 12, where A, B, H Jy H B, the 8^, and the y i are as before. I(a). First, we suppose that our odd cut is proper but that S=o (ft - *)(fy -1) ^ 1 and that neither H A nor H B is the exceptional graph of this theorem. Then i=0 i=0 i=0 and we have done. I(b). Next suppose that one of H A and H B is the exceptional graph, say H A, and the other, H B, is not. Then H has the appearance of Fig. 19. H- Now <p(h A ) = 7; hence FIG. 19 <P(H)= i i=0 i=o i=0 i=0 and again we have finished.

15 174 L. LOVlSZ AND M. D. PLUMMER I(c). Thus we may suppose that both H A and H B are the exceptional graph, in which case H is one of the two 18-point graphs of Fig. 20 and the reader may easily check that each contains more than 12 = p 1-factors. FIG. 20 II. Then we may suppose that for all proper odd cuts, i=o Hence by Lemma 4.1, H is one of the graphs of Fig. 13. However, the graph of Fig. 13(a) has p = 10 and f (p 1) = 6 = <p{h). Thus the graph of Fig. 13(b) is the only exception.* III. Thus we may assume that H has no proper odd cut and, furthermore, we may assume that G is not a wheel, for the theorem clearly holds for wheels. Thus H contains 2-fans. Form a new graph H' by contracting all 2-fans of H to a single point. If k denotes the number of 2-fans contracted, H' has p' = p 2k points. We claim that H' e 2&. For let C be any non-trivial cut of H'. Then if C is the corresponding cut of H, C is a proper cut of H and is thus even by assumption; and the claim is verified. So, by Theorem 3.4, <p(h) ^ p' l =p 2k 1. Now recall that any 1-factor of H' produces a 1-factor of H. Let D be any triangle of H which was contracted to produce H', and form another new graph H" from // by contracting all 2-fans except D. Now H" has a 1-factor F" D containing no line of D because, if one removes two points adjacent to D but outside D, then a 1-factor of the remaining graph can be extended to such a 1-factor. F" D extends to a 1-factor F D of H which contains exactly one line of each contracted triangle of H other than D. There are k such different 1-factors of H, so <p(h) ^ p-2k-l + k= p-k-1. Moreover, k ^ %{p -1) unless

16 ON A FAMILY OF PLANAR BICRITICAL GRAPHS 175 H is the triangular prism shown in Fig. 21. However, this graph has p = 6 and <p{h) = 5 > ${p-l). FIG. 21 Thus we may assume that k «$ \{p 1); then This result is best possible in the sense that there is an infinite family of even Halin graphs where equality holds. They are constructed as follows: let W r be a wheel with r points on the rim each joined to a common point v, where r is odd. For each point on the rim insert a triangle (cf. Fig. 22). W' r : r triangles (r odd) FIG. 22 Then the resulting graph W' r has 3r+l points and 2r 1-factors. Thus 5. Epilogue One final reason for our interest in Halin graphs arises from the following CONJECTURE. Every ^-connected planar triangulation contains a spanning Halin subgraph. If this conjecture were true, Whitney's theorem ([9]) guaranteeing a hamiltonian cycle in such graphs would follow immediately. The generalization to 4-connected graphs which would correspond to Tutte's generalization ([8]) of Whitney's theorem is not true, as shown by the line-graph of the dodecahedron (found by Malkevitch to show that 4-connected planar graphs need not contain a 4-cycle).

17 176 ON A FAMILY OF PLANAR BICRITICAL GRAPHS REFERENCES 1. L. W. BEINEKE and M. D. PLUMMER, 'On the 1-factors of a non-separable graph', J. Combinatorial Theory 2 (1967) R. HALEST, 'Studies on minimally n-connected graphs', Combinatorial mathematics and its applications, edited by D. J. A. Welsh (Academic Press, New York, 1971), pp A. KOTZIG, 'Ein Beitrag zur Theorie der endlichen Graphen mit linearen Faktoren, I, II, III', Mat. Casopis Sloven. Akad. Vied 9 (1959) 73-91, , and 10 (1960) (in Slovak, with German summary). 4. L. LovAsz, 'On the structure of factorizable graphs', Acta Math. Acad. Sci. Hungar. 23 (1972) and M. D. PLUMMER, 'On bicritical graphs', Proceedings of the colloquium on finite and infinite sets, Soc. Janos Bolyai, 1973, to appear. 6. H. RADEMACHER, 'On the number of certain types of polyhedra', Illinois J. Math. 9 (1965) W. T. TUTTE, 'The factorization of linear graphs', J. London Math. Soc. 22 (1947) 'A theorem on planar graphs', Trans. Amer. Math. Soc. 82 (1956) H. WHITNEY, 'A theorem on graphs', Ann. of Math. 32 (1931) J. ZAKS, 'On the 1-factors of n-connected graphs', Combinatorial structures and their applications (Gordon and Breach, New York, 1970), pp Eotvos L. University Budapest Hungary Vanderbilt University Nashville Tennessee