10
The Importance of Being Euler Von H~,RBERT FLEISCH~ER*) In this paper it will be shown that a well-known conjecture on the existence of Hamiltonian cycles in a class of planar, cubic graphs can be transformed into an equivalent conjecture on the existence of a special type of Eulerian trails in their duals. The latter conjecture will then be generalized. If not defined otherwise, the concepts used can be found in [2]. The conjecture we start from is generally attributed to D. BARNETTE (U. C. at Davis). Conjecture A. Let G be a plane, three-connected, cubic, bipartite graph. Then G has a Hamiltonian cycle. It is a well-known fact that the deletion of bipartiteness in the above conjecture yields the famous (but false) conjecture by Tait which would have implied a positive answer to the Four-Color-Conjecture. The first counter-example to Tait's conjecture was given by W.T. TUTTE in 1946. Before we discuss Conjecture A we define two concepts. Definition 1. Let G be a plane, connected Eulerian graph. A Eulerian trail T : Vi, el, V2, e 2 .... , v n : v 1 01 G, (V(G) : {Vl, v2, . . .}, E(G) = {el, e~, . . .}), is called an A-trail i//or any section ei_l, vi, e i o/ T/ollows that ei and ei-1 are neighbors in the cyclic ordering o/the lines incident to vi. (For i -~ n we put ei = el.) In other words, in an A-trail any two consecutive lines belong to the boundary of a face of G. KOTZIG [3] calls--in the case of a plane, con- netted, regular graph of degree four--an A-trail a a-line. If we consider together with a plane, Eulerian graph G a fixed two- coloring (of the faces) of G, then we can relate Definition 1 to the following definition (we assume always the unbounded face to have color 1). Definition 2. Let v be a point o/ the two-colored, plane, Eulerian graph G such that deg v > 2. Let el, e2 ~, k ~_ 2, be the lines incident to v so that e2~-1 and e~i are part o/the boundary o/a/ace colored 1, e2i and e~i+l are part o/ the boundary o/ a /ace colored 2, 1 ~_ i g k, where we *) Supported in part by National Science Foundation grant GP-36418X.

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Page 1: The importance of being Euler

The Importance of Being Euler

Von H~,RBERT FLEISCH~ER*)

In this paper it will be shown that a well-known conjecture on the existence of Hamiltonian cycles in a class of planar, cubic graphs can be transformed into an equivalent conjecture on the existence of a special type of Eulerian trails in their duals. The latter conjecture will then be generalized.

I f not defined otherwise, the concepts used can be found in [2]. The conjecture we start from is generally attributed to D. BARNETTE (U. C. at Davis).

C o n j e c t u r e A. Let G be a plane, three-connected, cubic, bipartite graph. Then G has a Hamiltonian cycle.

I t is a well-known fact that the deletion of bipartiteness in the above conjecture yields the famous (but false) conjecture by Tait which would have implied a positive answer to the Four-Color-Conjecture. The first counter-example to Tait's conjecture was given by W.T. TUTTE in 1946.

Before we discuss Conjecture A we define two concepts.

Definition 1. Let G be a plane, connected Eulerian graph. A Eulerian

trail T : V i , e l , V 2 , e 2 . . . . , v n : v 1 01 G, ( V ( G ) : {Vl , v2 , . . .},

E(G) = {el, e~, . . .}), is called an A-trail i / / o r any section ei_l, vi, e i o/

T/o l lows that ei and ei-1 are neighbors in the cyclic ordering o / the lines incident to vi. (For i -~ n we put ei = el.)

In other words, in an A-trail any two consecutive lines belong to the boundary of a face of G. KOTZIG [3] cal ls-- in the case of a plane, con- netted, regular graph of degree f o u r - - a n A-trail a a-line.

I f we consider together with a plane, Eulerian graph G a fixed two- coloring (of the faces) of G, then we can relate Definition 1 to the following definition (we assume always the unbounded face to have color 1).

Definition 2. Let v be a point o/ the two-colored, plane, Eulerian graph G such that deg v > 2. Let el, �9 �9 e2 ~, k ~_ 2, be the lines incident to v so that e2~-1 and e~i are part o/ the boundary o / a / a c e colored 1, e2i and e~i+l are part o/ the boundary o/ a /ace colored 2, 1 ~_ i g k, where we

*) Supported in part by National Science Foundation grant GP-36418X.

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The Importance of Being Euler 91

put 2 k ~ 1 = 1. Now we replace v with k points v 1 . . . . . v k so that e~ and e~i+l are incident to v ~. This operation (which replaces a point o/ degree 2k > 2 with k points o/ degree 2) is called the 1-splitting o/ v (in G). An- alogously, i / we let e~ ~-1 and e~i be incident to v ~, we speak o/ the 2-splitting o / v .

I f we view the points v ~ as (distinct points) v, then one sees imme- diately that the faces having color 2 remain unchanged, while the faces having color 1 whose boundaries contain v become one face after applying the 1-splitting in v. The analogy holds in the case of a 2-splitting.

Suppose now the 2-colored plane, Eulerian graph (7 has an A-trail T, and suppose for v e V (G) with the lines e l , . . . , e~, k ~ 2, incident to v (the subscripts of the e/s chosen corresponding to their cyclic order), that el, v, e, is a section of T. Then it follows easily from the definition of an A-trail and the planarity of G that T induces in v a 1-splitting or 2-splitting (depending on the 2-coloring of G in v). T h u s - - f o r a given fixed A-trail T - - T induces a (unique) decomposition of Va(G)= {v e V (G) ] deg v > 2} into two sets S~ and $2 where S~ contains all points in which T induces a 0-splitting, 0 =- 1, 2. (In this general form, one of the sets may be empty.) In the following considerations and because an A-trail can be defined for plane multigraphs as well, we shall always assume that Vg (G) = V(G). One sees easily that in this case we have a partition {S1, $2} of V(G) induced by an A-trail, i. e., $1 ~= r S~ ~= 0.

Definition 3. Let G be a 2-colored, plane, Eulerian graph; let C = (FI, . . . . F,~; m > 1} be a set o/ /aces belonging to the same color-class, and let bdFi denote the boundary o/ F i . Suppose there exist distinct points vl~, �9 �9 V,~l such that v~, i+l ~ bdFi n bdF~+l, i = 1 . . . . . m, putting m ~ 1 -= 1. Then we call C a unicolored/ace-ring and Lc -= {vl~, . . . , vmi} a complete set o/ links o/ C.

I t is clear that for a given unicolored face-ring C, there may exist different complete sets of links. However, if G is 3-connected, then such a ring C defines a uniquely determined Lc.

We are now going to state an obvious but useful lemma. There and later on in the proof of Theorem 1 as well, Gv denotes the graph obtained from G by applying the corresponding 0-splitting to the points of G if G has an A-trail which induces a partition P---- {$1, S~} of V (G).

Lemma 1. The two-colored, plane, Eulerian graph G has an A-trail i / and only i / there exists a partition P-= {S~, $2} o/ V (G) so that alter ap- plying the O-splitting to all v ~ S~, O ---- 1, 2, the resulting graph Gp is a cycle.

The following criterion relates Definition 1 to the Definitions 2 and 3.

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92 Herbert Fleischner

Theorem 1. Let G be a two-colored, plane, two-connected, Eulerian graph. Then G has an A-trail i/ and only i/ V (G) can be partitioned into (non- empty) classes S~, $2 such that /or any unicolored lace-ring C whose ele- ments have color 6, and any complete set o/ links Lc /olIows Lc r S~, 6e{1, 2}.

P r o o f . Suppose G having an A-trail T. From the above considerations we know that S~, the set of all points in which T induces a 6-splitting, is non-empty for 3 ---- 1, 2, and because of our general assumption ($1, $2} is a partition of V (G). Now let C be a unicolored face-ring with its ele- ments having color 6, and L c a complete set of links of C. Suppose v = vi.i+l belonging to bdFi n bdFi+l. After applying the 6-splitting to v we obtain a graph having a face F with E (bdF) D E (bdFi) u E (bdF~+l). Thus it is clear from the fact that Lc contains at least two distinct ele- ments, that Gp is disconnected if Lc belongs entirely to S~. This con- tradicts Lemma 1. Thus the condition as stated in Theorem 1 is necessary for G having an A-trail.

Conversely suppose S 1 and S 2 forming a partition of V(G) such that every complete set of links, Le , of an (arbitrary) unicolored face-ring C fulfills L c r Sz if the elements of C have color 6. Applying to the elements of S~ a 6-splitting we obtain the graph Gp which is a cycle, or each component of Gp is a cycle. By Lemma 1, if Gp is a cycle then P = {$1, S~} defines a (uniquely determined) A-trail of G. Thus we con- sider the case that Gp is disconnected. So we can find a minimal subset So of V(G) so that if we apply to each v 0 ~ So with v0 ~ S~ (6 -~ 1, 2) the 6-splitting, then the resulting graph G(So) is disconnected. Since G is two-connected S O contains at least two elements. Without loss of generali- t y the unbounded face F0 of G(S0) has a disconnected boundary. As- suming F0 having color 1 and ff w e S O belongs to $2, then G(S o --(w}) has already a disconnected boundary for its unbounded face. Since this contradicts the minimality of S O we have So C S 1. I t now follows easily from the minimality of So that the faces of G whose boundaries form bdFo, contain the elements of a unicolored face-ring C of G, and that So is a complete set of links of C. Since F 0 has color 1 and S O C $1, the con- tradiction to the hypothesis is shown. This finishes the proof of Theorem 1.

I f we consider a partition P ---- ($1, $2} as stated in Theorem 1, and if we apply to the elements of S~ a (6 q- 1)-splitting (putting 3 ---- 1) then the resulting graph may not be a cycle, i. e., viewed the elements of P the other way around, P may not define an A-trail of G. In other words, ff we exchange in the two-coloring of G the color classes, then, in general, we also have to exchange the meaning of the classes S 1 and $2. In the following we say that a partition of V (G) into two classes, P ~ (A, B},

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The Importance of Being Euler 93

is an A-partition if there are A-trails N1 and N 2 of G such that N1 in- duces the 1-splitting for the elements of A and the 2-splitting for the elements of B, and N2 induces the 2-splitting for the elements of A and the 1-splitting for the elements of B. Thus an A-partition yields an A-trail of G independently from the actual two-coloring of G. We also note tha t the definition of a unicolored face-ring of G is not depending on the actual two-coloring of G.

Corollary 1. Let G be a (two-colored), plane, two-connected, Eulerian graph. Then a partition P = (A, B) o/ V (G) is an A-partition i/ and only 4/ /or any complete set o/ links Lc o/ any unicolored ~ace-ring C /ollows L c r A, L e t B.

P r o o f . We prove Corollary 1 by twofold application of Theorem 1. I f P ~-- (A, B} is an A-partition, then there are A-trails N 1 and N 2

such that $1 ~-- A, $2 = B for N~ and S~ = B, S~ ---- A for N 2. Assuming now G to be two-colored we find by Theorem 1 for any complete set of links Lc of any unicolored face-ring C whose elements have color 8, that Lc r S~. But S~ is in one case equal to A and in the other case equal to B. Thus Lc r A, L c r B.

On the other hand, independently cf whether we define S~ = A, S~ ---- B or $1 ---- B, S~ ---- A, the partition P ---- (A, B} = {$1, S~} fulfills the condition of Theorem 1. Thus we find A-trails N1, N2 which have the required porperty, and therefore, P ---- {A, B} is an A-partibion. This finishes the proof of Corollary 1.

Let us now consider a plane triangular graph G satisfying the hypo- thesis of Theorem 1. I f we consider here a unicolored face-ring C whose elements have color 5 and a corresponding complete set of links Lc, then the following two statements are obviously true:

1) (Lc}, the subgraph of G induced by Lc, contains a cycle.

2) Every line of such cycle is boundary line of a face colored 5 -t- 1 (we put 3 = 1). Thus, unless these lines belong all to the same boundary, we find a unieolored face-ring C' whose elements have color 5 ~- 1 with a complete set ef links Lc, and Lc, C Lc.

From these two statements and Theorem 1 the following corollary is immediate.

Corollary 2. Let D be a two-colored Eulerian triangulation o/ the pla~e. Then D has an A-trail T 4/and only i/ V (D) can be partitioned into two classes V~ and V2 such that i / (V~} contains a cycle K, then K is boundary o/ a /ace o/ D colored 8, 5--~ 1, 2.

On the other hand, a Eulerian triangulation of the plane D can be interpreted as the dual graph of a plane, three-connected, bipartite,

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94 Herbert Fleisehner

cubic graph G. I t is a well-known fact that G has a Hamiltonian cycle if and only if D --~ D (G) has a partition of its point set into classes D1, D, such that both (D1) and (D,) are trees (this is true also if we delete bipartiteness).

Definition 4. Let G be a plane, Eulerian graph having an A-trail T, and let F be a lace o I G. We say that T separates the lines el bdF i] no pair o/ (adjacent) lines o/ bdF appears in a section ei-1, vi, ei of T.

Looking at Corollary 2 we see tha t the existence of a cycle K ~- bdF in (V,) implies tha t T separates the lines of bdF (see also the inter- pretation of what a 5-splitting means). Thus we have immediately the following relation between Hamiltonian cycles and A-trails.

Theorem 2. A plane, three-connected, bipartite, cubic graph G has a Hamiltonian cycle if and only i] its dual D (G) has an A-trail which does not separate the boundary lines of any /ace of D(G).--Therefore, Con- jecture A is equivalent to the following conjecture.

C o n j e c t u r e A'. Every Eulerian triangulation of the plane, D, con- tains an A-trail which does not separate the boundary lines of any face of D.

However, the following conjecture A" will appear to be equivalent to the previously stated conjectures.

C o n j e c t u r e A". Every Eulerian triangulation of the plane contains an A-trail.

I t suffices to show that Conjecture A" implies Conjecture A. Let D be an arbitrary Eulerian triangulation of the plane. For every face F of

C

? /%

(1

\

b Figure 1

D with a, b, c being the boundary points of F, we embed in F a copy of the graph of Figure 1 identifying the points having the same label. Thus we obtain a Eulerian triangulation of the plane D' with D C D'. Assuming now that D' is two-colored and has an A-trail T', we find a

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The Importance of Being Euler 95

partition of V(D') into classes V'I and V', such that if (V~) contains a cycle K', then K' is boundary of a face of D' colored 6, ~ e {1, 2} (Corollary 2). In particular, no three points forming the boundary points of a face of D belong to the same class V~. On the other hand, this partition of V (D') induces a partition of V(D) into two (non-empty) classes V1 and V2, and by construction both (V~) and (V~) are acyclic. Bub this already implies for plane triangulations that these two induced subgraphs are trees. As formulated above, this means for the cubic graph G having D as its dual, that G is Hamiltonian.

Although we did not mention it in the formulation of Corollary 2, it is clear that (V~) is connected for ~ ~-- 1, 2 because of the triangular shape of each face of D. (More exactly, in a plane triangulation every separating point set induces a cycle which is not a boundary cycle).

In all so far unsuccessful attempts by the author to prove Conjecture A" the triangular shape of the faces of the graphs considered, did not appear as the really important property te deal with (the graphs are automatically assumed to be plane and Eulerian). What seemed to matter was the fact that the graphs in consideration are three-connected. This property is automatically included in a triangulation of the plane since we disregard graphs with multiple lines. This observation leads to the following conjecture which implies the preceding conjectures.

C o n j e c t u r e B. Every plane, three-connected, Eulerian graph con- tains an A-trail.

After we showed the relevance of A-trails in connection with an un- solved Hamiltonian problem we want to proceed to a question from which the author originally was led to the consideration of A-trails several years ago.

Let G be a plane, connected, cubic, bipartite graph. Such a graph has a unique face-coloring CF with three colors 1, 2, 3 from which we obtain a one-factorization of E(G) into factors L1, L2, L3 by letting e e E(G) belong to L , if e belongs to the intersection of the boundaries of faces colored i, ~" respectively, and if {i, ?', k}-~ {1, 2, 3). We call this one- factorization the "natural one-factorization" and write shortly NOF.

KOTZIO investigates and describes in a series of papers those cubic graphs which have a one-factorization L1, L3, L3 so that Li t3 L~ de- scribes a Hamiltonian cycle for i ~ ]. He proves in [4] that there is no plane, bipartite, cubic graph with such one-factorization. Thus one may rather look for a Hamiltonian cycle containing a given one-factor of a cubic, bipartite graph. Here we are concerned with Hamiltonian cycles in plane, connected, bipartite, cubic graphs G, and it seems natural to look for such Hamiltonian cycles which contain an Li of the NOF. To

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96 Herbert Fleischner

relate this question to A-trails we assume G also being three-connected and construct for i ---- 1, 2, 3 the plane, Eulerian (multi)graph G~ which we call the i th color reduction of G as follows: We contract each face F of G having color i and its boundary to a point p (F) of Gi; thus the lines of L~ become the lines of G~, and we require that the cyclic order of the lines incident to p(F) corresponds to the cyclic order in which these lines are incident to the points of b dF. Thus Gi is uniquely deter- mined by G for fixed i e {1, 2, 3}, and since G is three-connected G~ has no loops; but G~ possibly has multiple lines. We assume the definition of an A-trail extended for multigraphs. The following result can easily be proved by the reader.

Theorem 3. Let G be a plane, 3-connected, cubic, bipartite graph, and let L1, L~, L 3 be the NOF o/G. Then G has a Hamiltonian cycle containing Li i/ and only i/ the i th color reduction o/ G has an A-trail.

I t is clear how Conjecture B can be reformulated-- in view of Theorem 3 - - a s a conjecture on Hamiltonian cycles of a special type in plane, cubic, bipartite graphs. But Conjecture B implies a number of state- ments on several other classes of planar graphs. Let G denote a plane, 3-connected Eulerian graph. Then Conjecture B has its implications for the medial graph M(G) (see [5]), for the totM graph T(G), but also for a large class of bipartite graphs constructed from G.

From now on G always denotes a plane, Eulerian graph.

We noted above Kotzig's result that if G is regular of degree ibur, then G has an A-trail if and only if G is connected. That connectedness

Figure 2

does not suffice if G has a point of degree greater than four, can be seen from Figure 2. As simple as this figure may be, it characterizes the stage which one has to avoid if one starts from a two-colored, connected graph G and applies step by step 1-splittings and 2-splittings.

For some time the author thought that if G is assumed to be two- connected, then one can get around the "forbidden stage" as illustrated

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The I m p o r t a n c e of Being Euler 97

in Figure 2. However, from one of his (necessarily) unsuccessful at tempts to show this, arose the counter-example of Figure 3. We prove tha t this graph G has no A-trail. First we note the fact if a Eulerian graph has an A-trail and a cutpoint c, then for all A-trails the (S-splitting in c is the same. Now we consider the graph of Figure 3. Because of sym- metry reasons we apply to point 1 w. o. 1. o. g. a 1-splitting (the un- bounded face has color 1), thus obtaining the graph G1 for which point 2 is a cutpoint. Therefore we have to apply in 2 a 2-splitting, otherwise the resulting graph G2 would not be connected. Point 3 is a cutpoint of G 2 which implies that a 1-splitting has to be performed in 3, and so on. Thus the splittings in the points 2, 3, 4, 5, 6 are uniquely determined ff the &splitting in 1 is given. The graph G6 is still connected and has v as its only cutpoint. However, in Ge we have the situation as described in Figure 2. Therefore, we obtain from G 6 a disconnected graph after we apply in v either a 1- or a 2-splitting. Thus G contains no A-trail. Note the fact that G has only points of degree four and six.

Figure 3

For a further discussion of the graph of Figure 3 and a generalization of Conjecture B we need the following definition.

Definition 5. A subgraph H o/ a two-connected graph G is called a 2- component o/ G i]

1) H is two-connected;

2) exactly two points o/ H are incident to elements o~ E ( G ) - E(H),

3) I E ( G ) - - E ( H ) I > 1. 7 Hbg. )~ath. Abh., Bd. XLII

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98 Herbert Fleischner

I t is clear that i/ u (G) -~ 2 and i / G has no points o/degree two, then G has at least two 2-components. I t is also clear that a proper subgraph H' o/ a 2-component H can be a 2-component o/ H and there/ore o] G.

For each 2-component H of G there are exactly two faces F~, F2 of G which do not belong to H and for which E(bdF~) n E ( H ) ~ r i ---- 1, 2. Now suppose G to be two-colored. Depending on the coloring of F~, F2 we distinguish three types of 2-components:

Type 1) Both F~ and F~ have color 1.

Type 2) Both F~ and F~ have color 2.

Type 3) F1 and F 2 have different colors.

We note tha t the graph G in Figure 3 contains 2-components of all three types. But even if we consider only the minimal 2-components (i. e., those no proper subgraph of which is a 2-component of G), then we find among these such of Type 1 and Type 2. The author constructed a two-connected graph which contains no A-trail, and all 2-components of this graph are of Type 1 or Type 3; the minimal ones are only of Type 3. So the following question arises: I f G is two-connected, and ff every 2-component of G is of the same type, does then G have an A-trail? The author has no opinion on the answer to the question as far as it concerns Type 1 and Type 2. However, concerning Type 3 we believe tha t the answer is positive.

Finally, we want to point ou t - -w i thou t going into too many detai ls-- an interesting relation between A-trails and automorphisms in the case tha t G is three-connected.

Theorem 4. Let G be a plane, three-connected, Eulerian graph having an A-trail T ~ vl , el, v~, e 2 . . . . , and let o~ be an automorphism o/ G. Then T~ -~ o~vl, o~el, ocv~, o~e2 . . . . is an A-trail o/ G. Furthermore, H ( T ) = {or eF(G) I T~ : T /or fixed T} is a subgroup o/ I~(G), and H (A) : (o~ eF(G) I T~ -~ T / o r all A-trails T o I G} is a normal subgroup

ol

Proo f . I t has been proved by the author [1] tha t for each such G and any ~ eF(G), if el, e ~ , . . . , ea are the lines incident to the point v cor- responding to their cyclic order, then either ~el, a e ~ , . . . , sea or sea, aed_~, �9 �9 ~el describes the cyclic order of the lines incident to ~v (and either case holds simultaneously for all v e V(G)). Thus, if ei_l, v~, e~ is a section of T, then ei-1 and ei are neighbors in the cyclic order of the lines incident to v~, and so are ~e~_~ and ~e~ corresponding ~o avi. There- fore, T~ ~- ~ vl, ~ ex . . . . is an A-trail of G.

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The Importance of Being Euler 99

Because of the definition (~xfl)x -- a(flx) for x ~ V (G) u E(G) one sees

immedia te ly tha t T ( ~ : {T~)~. Thus we have for a, f l e H ( T ) or

o~, f l~ H(A) in any case T ( ~ : (Ta)~---- T~---- T (in the first case for

fixed T, in the second case for a rb i t ra ry T). I. e., also aft ~ H(T) , H(A) respectively, which suffices for H ( T ) and H(A) to be subgroups of

F(G) because of the finiteness of/~(G). Let a e H(A) and ~ eF(G) be

arbi t rar i ly chosen. Then T(y~r- b ---- (Tr-~)~y~ ---- ((T~-~)~)r ~-- (Tv-I)r = T(rr- b ----- T, i.e., 7 - ~ c ~ e H ( A ) . This finishes the proof of the theorem.

Bibl iography

[1] H. FI,EISCHI~ER, The uniquely embeddable planar graphs, Discrete Mathe- matics 4 (1973) 347 --358.

[2] F. HAI~Y, Graph Theory, Addison-Wesley, Reading, Mass., 1971. [3] A. KoTzIo, Eulerian Lines in Finite 4-Valent Graphs and Their Transfor-

mations, Theory of Graphs, Proe. Colloq. Tihany, 1966 (P. Erd6s, G. Katona, editors), Academic Press, 1968.

[4] A. Kotzig, The Construction of the Harniltonian Graphs of Degree Three (Russian), (~as. p~st. mat. 87 (1962), 148--167.

[5] O. ORE, The Four-Color Problem, Academic Press, New York, London 1967.

Eingega~gen am 19. 9. 1973

Zusatz bei der Korrektur (18. 6. 74) :

Als sich die Arbeit im Endstadium der Drucklegung befand, entdeckten Frl. S. R~.O.~ER und der Autor, dab die Vermutung B falsch ist. In der Tat kann man beweisen, dab aus der Giiltigkeit von Vermutung B folgen wiirde, dab jeder ebene, ch'eifach zusammenh~ngende kubischo Graph eine Hamiltonsche Linie besitzt.Wio jedoch 1946 von W. T. T~-~rE und sparer yon anderen gezeigt wurde, ist diese Annahme falsch. Daher muB auch Vermutung B falsch sein.

Professor HERBERT FLEISCHNER

Institut fiir Informationsverarbeitung

Osterreichischo Akademie der Wissenschaften

Fleischmarkt 20/I/4

A-1010 Wien, Austria

7*