Abh. Math. Sem. Univ. Hamburg 64 (t994), 141-150

One-way Infinite Hamiltonian Paths in Infinite Maximal Planar Graphs

By H.-O. JUNG

Abstract. A one-way infinite Hamiltonian path is constructed in an infinite 4-con- nected VAP-free maximal planar graph containing one or two vertices of infinite degree. Combining this result and that of R. HALXN who investigated the structure of such graphs, we conclude that such a path always exists in every infinite 4-connected maximal planar graph with exactly one end, which is an extension of H. WmTNEY'S theorem to infinite graphs.

I Introduction

In finite graphs, the existence of Hamiltonian cycles in 4-connected maximal planar graphs was verified by Wm'ryEY [7], and later TtrrTE [6] and TnOMASSEN [5] generalized this property to all 4-connected planar graphs. Our interest arises from the following conjecture which is an extension of TtrrrE's theorem to infinite graphs:

Conjecture. (NASH-WILLIAMS [3]) Every infinite 4-connected planar graph with exactly one end has a one-way infinite Hamiltonian path.

This conjecture is yet open and seems to be considerably complicated. In [2], JUNG showed that every 4-connected infinite VAP-free maximal planar graph without vertices of infinite degree (such a graph is called a 4-connected "strong triangulation", following Tt~OMASS~N [4]) has a one-way infinite Hamiltonian path. This theorem is a part of an extension of WmTNEY'S theorem, and in addition it took a first step to reach a solution of the NAsH-WILLIAMS' conjecture.

In this paper, we will extend the result of JUNG to general maximal planar graphs under the same hypothesis. Namely we improve his theorem as follows:

Theorem. Every infinite 4-connected maximal planar graph with exactly one end has a one-way infinite Hamiltonian path.

It is clear that any infinite graph must have exactly one end to contain a one-way infinite Hamiltonian path. In particular, if the graph is maximally planar this property is equivalent to have a VAP-free representation, which is investigated by HALIN [1] and is stated below. It is noted that the property above is again equivalent to have a 2-basis, by TtJOMASS~N [4]. Therefore, from HALIN'S result, it is sufficient to restrict to graphs having a VAP-free representation.

142 H.-O. Jung

Lemma. (HALIN [1]) An infinite maximal planar graph contains exactly one end if and only i f it has a VAP-free representation. In this case, the graph is 3-connected, contains at most 2 vertices o f infinite degree and these vertices must be adjacent to each other.

Let G be an infinite VAP-free maximal planar graph. If G contains no vertices of infinite degree, then, as mentioned above, the existence of a one-way infinite Hamiltonian path is already verified. Therefore, in oder to investigate our main result, it is sufficient to prove the following theorem. In particular, our proof is constructive as that in [2].

Theorem. In every 4-connected VAP-free maximal planar graph containing one or two vertices o f infinite degree, there exists a one-way infinite Hamiltonian path.

We shall, for the whole paper, use the terminology defined in [2], except for following notations:

The set of all integers is denoted by Z. While the set of all natural numbers including 0 is denoted by No, we let N := {1,2,3,...}.

2 Structure of VAP-free Maximal Planar Graphs

In this chapter, a VAP-free infinite maximal planar graphs containing one or two vertices of infinite degree will be characterized by the sequence of disjoint paths satisfying the suitable conditions, while their union covers all vertices of the graph. For this, we first inquire into some properties of such graphs.

We may say that a vertex u in an infinite connected plane graph G is one-way (resp. two-way) infinite, if there exists a bijective function

(*) ~0:N~(u) , No (resp. Z),

such that tp-l(n) and ~o-~(n + 1) are contained in a common facial cycle of G, for all n E No (resp. n E Z). If G is 2-connected and VAP-free, then every vertex of infinite degree in G is either one-way or two-way infinite. For, let u be a vertex of infinite degree in G. Since G - u is connected there exists at most one unbounded face incident to u in G, i.e., one facial path containing u in G. If G contains such a facial path we can easily see that u is one-way infinite, and otherwise u must be two-way infinite. Moerover, if G is maximally planar, then, in any case, ~p-l(n) is adjacent to q~-i (n + 1), since {u, ~o -1 (n), q~-I (n + 1)} must constitute a facial cycle containing no vertices in the interior.

Therefore we have:

Lemma 2.1. Every vertex of infinite degree in a VAP-free maximal planar graph is either one-way or two-way infinite.

Furthermore if u is a one-way (resp. two-way) infinite vertex with respect to the bijective function tp, then

{~-1 (0), q~-I (1), ~- t (2),...}

(resp. {.. . , tp -1 (--2), r (_ 1), ~o -1 (0), q~-I (1), q~-I (2), �9 �9 "})

One-way Infinite Hamiltonian Paths in Infinite Maximal Planar Graphs 143

is a one-way (resp. two-way) infinite path. []

Lemma 2.2. Let G be a VAP-free maximal planar graph.

(1) I f G contains exactly one vertex of infinite degree, then the vertex is two- way infinite.

(2) I f G contains exactly two vertices of infinite degree, then both are one-way infinite.

Proof (1) Assume that u is a vertex of infinite degree and is one-way infinite. From the definition, we then have a bijection q~ satisfying the property (*). Let v0 = q~-l(0), Vl = q~-l(1), and let N6(vo) = {Vl,U, Wl ," ",wr} in natural order. Note that the degree of v0 is finite and > 3. Since G has no accumulation points and there can be no more vertices adjacent to v0 between u and Wl, the vertices u, v0 and Wl are contained in a commom facial cycle or facial path. From the maximal planarity we conclude that UWl E E(G), which contradicts our construction of q~.

(2) Let u- and u + be the vertices of infinite degree in G, and assume that at least one of two vertices is two-way infinite, w.l.o.g, u- is such a vertex. Then, as is defined in (*), we have a bijective function

q~: N~(u-) , Z

holding the corresponding properties. By Lemma 2.1, W := q~-l(Z) = No(u -1) is two-way infinite path. Let v0 := q~-i (0) and let W + and W - be the one-way infinite subpath of W originating v0 containing qg-l(1), q~-l(-1), respectly. That is, W - and W + are two one-way infinite subpaths of W such that

W = W - U W + and W - N W+ = {vo}.

Finally choose an arbitrary vertex w(~ v0) adjacent to u +. From the 3-con- nectedness of G, G - {u-,u +} is connected, and so there is a path P of G - {u-,u +} connecting v0 and w. Since u-u + E E(G), by HALIN'S Lemma in chapter l, we have a cycle P U {you ,u u +,u+w} of G. Then one of two paths W - -- v0 and W + - v0 must lie in the interior of the cycle. For, assume that all vertices of W (up to the vertex vo) lie in the exterior of the cycle. Then, since G is VAP-free, the paths W - and W + must be unbounded, and therefore either the vertices of W - - v0 or W + - v0 cannot be adjacent to u-, which is impossible.

Since not only W - - v o but also W + - v0 contains infinitely many vertices, it follows that G must have an accumulation point in the interior of the cycle, which contradicts our assumption. []

Now let G be of the case (2) in Lemma 2.2. If G is separated by 3 vertices, then it can be easily seen that the paths W - and W + are not always disjoint, i.e., W cannot always be a path, and further they are not always induced, where W-, W + and W are defined in the proof in the Lemma 2.2. But, if G is 4-connected, then we show the following:

144 H.-O. Jung

Corollary 2.3. Let G be a VAP-free 4-connected maximal planar graph with two vertices u- and u + o f infinite degree. Then G - {u-, u +} has exactly one facial path W. Further, there exists a vertex vo E V(W) such that all vertices of one one-way infinite subpath o f W originating vo must be adjacent to u- and those of the another adjacent to u +, and moreover these two paths are induced.

Proof. Let v0 be the vertex contained in the facial 3-cycle which contains the vertices u- and u+; note that u- and u + are adjacent to each other, by

�9 HALIN'S Lemma in chapter 1. By Lemma 2.2, the u- and u + are one-way infinite, and so we have two one-way infinite paths W - and W+, whose vertices are adjacent to u- and u +, respectly, by Lemma 2.1. Then it can be easily seen that u + (resp. u-) is the originated vertex and v0 is the next following of W - (resp. W+). Let W - := W - - u + and W + := W+ - u-. Then, by setting W := W - U W +, we get a two-way infinite facial path of G - {u-,u +} satisfying the desired properties. The paths W - and W + must clearly be induced from the connectivity number. That W is the unique facial path of G - { u - , u +} follows from the hypothesis and the fact that G - { u - , u + } has no vertices of infinite degree. []

It will be helpful to introduce some notations which will be used throughout this paper: Let G be a VAP-free 4-connected maximal planar graph. If G contains only one vertex u of infinite degree, then u is a two-way infinite vertex, by Lemma 2.2. So, by Lemma 2.1, we have a two-way infinite facial path W of G - u

W = { . . . , q~-I ( -2 ) , ~o -1 ( -1 ) , (p-1 (0), q~-I (1), ~o -1 (2 ) , . . . } .

Then, we set

W - = { ( p - l ( o ) , t p - l ( - 1 ) , t p - l ( - 2 ) ' " } ,

W + : {tp-l(o),r.p-l(1),(p-l(2) "' '} and

v0 = ~o-1(0).

On the other hand, if G contains two vertices u- and u + of infinite degree we will in this case use the same notations as in Corollary 2.3 and its proof, such as v0, W - and W +.

Then, in each case, it must hold:

W = W - U W + and W - n W+ = {vo}.

We are now prepared to define several conditions: A finite path P of H := G - u (or H := G - {u-,u+}) is a W-path of G, if one of its endvertices lies on W - and the another on W +, and additionally the remaining vertices of P do not lie on W. Note that such paths are nontrivial, since W is a two-way infinite path. Note also that, for a W-path P with endvertices v- and v +,

P U (vo, v--path on W-) U (vo, v+-path on W +)

One-way Infinite Hamiltonian Paths in Infinite Maximal Planar Graphs 145

builds a cycle, which will be denoted by Cp. For disjoint two W-paths P and U, we may define a (P, P')-fan as a subgraph of H, which consists of not only P and P ' but also the vertices and edges lying between P and P'. A chord of a (P, U)-fan R is an edge joining a vertex of P and that of pr, and next Wk- and W + denote the paths on W - and W +, respectly, connecting an endvertex of P and that of P'. Note that if W~- or W + contains only one edge it is simultaneously a chord of R.

We now let B be the set of all chords of R, and set

:= (P U P')U BU WR U W+R .

For an inner facial cycle J of B, we define a cell of R as the subgraph 7 of G which contains all vertices and edges not only on J but also in the interior of J. If J = J, then the cell is called empty. A (P, P')-fan is normal if it satisfies following properties:

[N1] P and P ' are induced.

[N2] Every vertex of P ' is adjacent to a vertex of P.

Remark. If (P,P')-fan is normal, then we can easily verify that:

(1) 1 < IV(L) n V(P')I < 2, for every cell L of R.

(2) If V(L) n V(P') = {z,z'}, z ~ z', for a cell L, it must hold zz' E E(P').

From the property (1) in Remark above, the set V(L) n V(P') contains either one or two vertices for every cell L. As is already defined in [2], we may say that L is of type 1 in the former case and of type 2 in the latter case.

Lemma 2.4. Let P be an induced W-path of a VAP-free 4-connected maximal planar graph G containing a vertex of infinite degree. Then there exists a W- path P' disjoint to P, such that the resulting (P, P')-fan is normal.

Proof Set H := G - u or H := G - {u-,u+}, if G contains exactly one or two vertices of infinite degree, respectly. We will first construct a W-path P' of H. Since P is an induced W-path of H, we have two induced subgraphs H1 and HE such that

HI U H2 = H, H 1 n H2 = P and vo E V(H1).

Then let F := {J [ J is an inner facial cycle of H2 satisfying V(J) n V(P) ~ 0} and let E be the set of all vertices of the cycles of F. Note that hE1 < ~ , since every cycle of F is finite and F is also finite set. We further define 0- as the vertex in V(W-) N E such that the metric distance between 0- and v-, i.e., d(v-, i)-), has maximum for all elements in this set. Let/~+ be similarly defined in V(W +) N E. Finally we denote by K the induced subgraph of H2 (and also of H) containing all elements of E and all vertices of the v , v -path on W -

146 H.-O. Jung

and the v +, b+-path on W +. Then we can easily see that K is a 2-connected finite subgraph of H2 and that the path P is contained in the outer cycle of K. By setting P ' as the b-, b+-path on the outer cycle of K which doesn't contain a vertex of P, we will show that the resulting (P,P')-fan satisfies the conditions [N1] and [N2].

First, from the 2-connectedness of K, it follows that the paths P and P ' are disjoint, since these paths lie on the outer cycle of K. As K is an induced subgraph of H, P ' is also an induced path. By considering of the fact that all inner facial cycles of K are triangles, it can be easily verified that the assertion [N2] holds from the construction of F (and also K). []

With aid of Lemma 2.4, we are now ready to establish the main result discussed at the beginning of this chapter.

Proposition 2.5. Let G be a VAP-free 4-connected maximal planar graph containing a vertex of infinite degree, and let u (or u- and u+), vo, W and Cp be defined above. Further let H = G - u or H = G - {u-, u+}, and let P1 denote the W-path consisting of the vertices adjacent to vo. Then there exists a sequence of induced W-paths {P1, P2, P3,'" "} satisfying the following properties:

(1) The (Pj, Pj+l)-fan is normal, for all j E N.

(2) V(H) = v ( U j = 1 -Cpj).

Proof It is clear that P1 is an induced path from the fact of 4-connectedness of G. For j E N, the existence of Pj+I, related to Pj, satisfying the condition (1) follows from Lemma 2.4. It remains only to show that the resulting paths {PI,P2,'" "} hold the condition (2).

Let x E V(H) be an arbitrary vertex. Since d(y,z) > 1 for all y ~ V(Pj) and z E V(Pj+I) (j E ~q), it follows that x ~ V(Cp, x ), where nx is a metric

O(3 distence between x and v0. Because of V(-Cpox ) c v(Uj=l-Cpj), we get

_ _ V o o - - V(G) = (Uj=l CPs). Since it holds clearly that V(G) ~ v(u~= 1 Cpj), we

can conclude V(G) = v(u~= 1 -Cpj). []

Remark. It may be mentioned without proof that, for the fixed vertex v0 on IV, such a sequence of induced cycles with the conditions (1) and (2) is unique.

3 Proof of the Main Theorem

Let G be a VAP-free 4-connected maximal planar graph containing a vertex of infinite degree, and let R be a normal (P, P')-fan in G. Further set H := G - u or H := G - {u-, u+}, if G contains exactly one vertex u or two vertices u-, u + of infinite degree, respectly. In order to prove our main theorem, we will, as mentioned, use the same notations as is defined in chapter 2. Additionally let v- (resp. v +) denote the endvertex of P on W - (resp. W+), and let b- (resp. ~+) denote that of P ' on W - (resp. W+).

One-way Infinite Hamiltonian Paths in Infinite Maximal Planar Graphs 147

If R contains at least 2 cells, according to the definition, there exist exactly two of them containing an edge of W, one of which, denoted by L- , contains an edge of W - and the another, denoted by L +, that of W +. On the other hand, if R has only one cell we set this by L +.

Now we first investigate the cells L - and L +.

Lemma 3.1. Let L - and L + be of type 1, and let v- and x - (resp. x + and v +) be the endvertices of the subpath of P intersecting to L - (resp. L+). Then there exists a v , x -path covering all vertices of L - - b - , and an x +, b+-path covering those of L +.

Proof To obtain the first part of the theorem, recall the path WR, which is the subpath of W - connecting v- and b-. It must also be noted that, as is constructed in chapter 2, W~- and the v , x -subpath of P are induced. Let z denote the vertex on WR adjacent to /~-, and set L := L - - / J - . From the definition of L- , b- cannot be adjacent to a vertex of P, up to v- and x-, and therefore L is 2-connected. We will now show that (L, v , x ,z) satisfies the condition W2. Then, by Lemma 4 in [2], we get a v , x path covering all vertices of L - - b - . As W R is an induced path, its v-, z-subpath is also induced on the outer cycle, say C, of L. Since the property for the v , x -path is already verified, it remains to show that the x- , z-path, denoted by J, is also induced on C. For this, suppose there exist vertices x,y E V(J) with xy E E (L)\ E (J). Then, as in the proof of Lemma 5 in [2], {x, y, i)-} constitutes a separating triangle, which implies a contradition.

Now observe the second part. By a similar argument as in the first part, we conclude that x +, v+-path and v +,/J+-path on the outer cycle of L + is induced. Since x+~ + is an edge of the cycle, it follows that (L +, x +, b +, v +) holds the property W2, and therefore we also find a x +, b+-path which covers all vertices of L +. []

Lemma 3.2. I f L - or L + be of type 2, then v-b- E E ( W - ) or v+b + E E(W+), respectly.

Proof. We will only prove the Lemma for the cell L- . By a similar argument, it can be also verified for L +. If V(L-) N V(P) = {v-},/~- must be adjacent to v- by [N2], and so, from the inducedness of W- , we get v-0- E E(W-) .

Now let IV(L-) n V(P)I > 2, and suppose dw-(v-,i~-) > 2. Then, since/~- must also be adjacent to a vertex of P by [N2], L - will be decomposed into two cells, and this contradicts our definition of L- . Therefore {v-, b-} must be an edge of W-. []

We will now construct a v-, b+-path which covers all vertices of R - (V(P') \ {b+}) (see Figure).

(1) The cell L - For this cell, we will construct a v , x -path PL which covers all vertices

of L- , except for those of P', where x - is defined in Lemma 3.1.

148 H.-O. Jung

First, if L - is of type 1, we get a v ,x -path satisfying the desired property, according to Lemma 3.1. Now let L - is of type 2, and set {b-, y-} = V(P') A V(L+). If v- = x- , then v-/)- is an edge of W - by Lemma 3.2, and so L - must be empty. In this case let PL := {v-}. Otherwise L - clearly is 3-connected, by Lemma 3.2, and contains no separating triangles. Combining this property and the fact that b-y- and x - y - are edges of the outer cycle of L- , L - satisfies the hypothesis of Lemma 5 in [2], and so we also have a v , x -path which covers all vertices of L - - {b-, y-}.

(2) L is a cell of R, except for L - and L + Let x and x' be the endvertices of the subpath of P intersecting to L.

Then, as in the case L- , we construct an xx'-path PL covering all vertices of L, except for those of P'.

Case 1: L is of type 1: Let first y E V(L) tq V(P'). If L is empty, then xx' must be an edge, and

in this case set PL := {xx'}. Otherwise L clearly is a 3-connected triangular graph without separating triangles. Since x and x' are adjacent to y on the outer cycle of L, we can find a x, x'-path PL which covers all vertices of L - y, by Lemma 5 in [2].

Case 2: L is of type 2: Let y, y' E V(L) f3 V(P'). If x = x' we set PL := {x}. Otherwise, since

yy' E E(P') and xy, x'y' E E(G), L satisfies the hypothesis of Lemma 5 in [2], so that we can also find an xx'-path covering all vertices of L - {y, y'}.

(3) The cell L + Let x + be the vertex defined in Lemma 3.2. For this cell, we will construct

a x +,/)+-path, which covers the vertices of L + or L + - y+, if L + is of type 1 or of type 2, respectly, where y+ is the vertex adjacent to x + and/)+ on the outer cycle of L +, i.e., {y+,/)+} = V(L +) N V(P').

In the case L + is of type 1, we can immediately find a path PL connecting x + and b + which covers all vertices of L +, by Lemma 3.1. On the other hand, let L + be of type 2. If x + = v +, let PL := {v+b+}. Otherwise, by similar argument to the case L- , L + satisfies the hypothesis of Lemma 5 in [2], so that we get a x +,/)+-path which covers all vertices of L + - y+.

+.

One-way Infinite Hamiltonian Paths in Infinite Maximal Planar Graphs

We now collect all paths constructed above:

149

~R := U PL. L: cell of R

T h e n ~i~ R clearly is a v-, b+-path of R, which covers all vertices of R - (V(P') \ {b+}), as desired. We summerize this result in a theorem.

Proposition 3.3. Let R be a normal (P, P')-fan, and let v-, v +,/~- and i; + are defined at the beginning of this section. Then there exists a v-, b+-path of R covering all vertices of R - (V(P') \ {b+}). []

Remark. By interchanging the symbols % ' and ,_,, we can also obtain a v +, b--path which covers all vertices of R -- (V(U) \ {~-}).

We are now prepared to prove the main result of this paper. Let G, u (or u- and u+), H, v0 and P1 be introduced in chapter 2 and in Proposition 2.5. Then we have a sequence of induced W-path {P1,P2,P3,"'} holding the properties (1), (2) in Proposition 2.5. For j E {1,3,5,...}, since Rj is normal, we get .a path ~ j connecting the vertices vy and vf+ 1 which covers all vertices of Rj - (V(Pj+I) \ {v++l}), by Proposition 3.3. On the other hand,

for j E {2, 4, 6 , " .}, we also have a path ~ j connecting v + and vT+ 1 covering all vertices of R j - (V(Pj+O \ {v7+1}), following Remark above. Further

let ~0 := {V0Vl+}. Then ~ ' := U j=0 j clearly is a one-way infinite path

originating v0. Because of V(H) = v(U~= I Cej), it must hold: V(~') = V(H), and therefore ~ is a Hamiltonian path in H. Finally set ~ := ~ ' U {u, v0} or

:= ~ ' U {u-u +, u+v0}, if G contains exactly one or two vertices of infinite degree, respectly. Then ~ is a one-way infinite Hamiltonian path in G, and this completes the proof of our main theorem.

References

[1] R. HALIN. Some Problems and Results on Infinite Graphs. Ann. Disc. Math. 41 (1989), 195-210.

[2] H.O. JUNG. An Extension of Whitney's Theorem to Infinite Strong Triangulations. Abh. Math. Sem. Univ. Hamburg 64 (1994), ??%???.

[3] C.ST.J.A. NASH-WILLIAMS. Unexplored and Semi-explored Territories in Graph Theory. New Directions in Graph Theory (ed. E Harary), Acad. Press, New York 1973, 149-186.

[4] C. THOMASSEN. Planarity and Duality of Finite and Infinite Graphs. J. Comb. Th. (B) 29 (1980), 244-271.

[5] C. THOMASSEN. A Theorem on Paths in Planar Graphs. J. Graph Th. 7 (1983), 137-160.

[6] W.T. "RrrrE. Bridges and Hamiltonian Circuits in Planar Graphs. Aequations Math. 15 (1977), 1-3.

[7] H. WmTNEY. A Theorem on Graphs. Ann. Math. 32 (1931), 378-390.

150 H.-O. Jung

Eingegangen am: 24.11.1992 in revidierter Fassung am: 17.03.1993

Author's address: Hwan-Ok Jung, Department of Mathematics, Han-Shin University, Osan-shi, Kyungki-do, 447-791, South Korea.