Planar and infinite hypohamiltonian and hypotraceable graphs

Planar and infinite hypohamiltonian and hypotraceable graphs

Discrete Mathematics 14 (1976) 337-389. Q North-HNand Publishing Company Received 25 April 1975 Revised f 2 September 1975 the question whether or no...

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Discrete Mathematics 14 (1976) 337-389. Q North-HNand Publishing Company

Received 25 April 1975 Revised f 2 September 1975 the question whether or not planar hypohamiitonian graphs exist and CXinbaum conjel;tuxed the nonexistence of such graphs. We shaHdescribe an infinite class sf pImar hy,poh;rmittaniangraphs and infMte classes of pladrarhypotraceable graphs of connectivity two (tesp. th.ree). Infinite hypohamiitonian (resp. hypotraceable) graphs are also described. ft is shown how the study of infinite hypotraceabk graphs leads to a new infinite famiiy af f”initehypotraceable graphs.

1. Intrduction

and terminology

A iinite graph CI’is hypohamiltonian (respectively hypotraceable) if it has no hamiltonian cycle (respectively path) but every ljsertex-deleted subgraph C - u hats. Analogously, we shall say that an enumerabiy infinite graph is hypshamiltonian (res ectively hypotraceable) if it has no 2-way infinite hamiltonian path (re txtively Lway infinite hamiltonian path) but wery veirtex-deleted subgraph has such a path. There exist several infinite classes of finite hypohamiltonia (for references, see, e. [ 141:) and in [ 131 it is shown how t hamj~tonian graphs. exists a finite

d ;afinite, cubk, non-plmar hypcdraw~blc rton’s camstmction so as to obtain iofinitely fimte, cthia= (mm-planar) ~~y~~~~ace~~ble graphs and infinitely marty -imnnected hy~tra~~a~~~~ graphs8 Van Diest [ 4 3 gtrve an exam,@e of an irlf’initt: hypohamik her or not m infinite hypcF e shall shm~ that them exist infinitefy many intr;acekble) gmphs and we shall the study of infinite hygotraceabk graphs kads to a new infinite f finite hygmtraceabfe graohs. is that of Iiar~~/ [ 101. Nowevu, we say ~mzx in7T-t~:tdge joining vertices x, y is ph C has ~mne~tiviry k and G is na? cocnhe;t G cmtains twa induced subgraphs GI, G, such tL?,t 1) 2, and V;IG,) f*f If($) =A i = k. 15% sirall s;iy that c, (and atso 4&) is a k-fragment of c1” ertic&v of a~ttachment o”fG, . In this paper we

irni

hypotmeable ce

of a finite, planar of Fig. 1.Ho hasIQ5ver-

e Into 66 Sganal rqiolrs and

. l(zt)- (h). The pknar hypohamibnian graphIi,

g)ne

~-gmtairegion.

Hence by the Ginberg criterion [63 (see, e-g*, 17, [ f 21). If0 is not hamiltonian. Becauie of the symmeconsider 13vertices in order to show 1 (a)! -. (h) show cycW (drawn with mg exactly ane vertex (wkich is marked) Of MO.In each --(h) a&~ another vertex is marked, since an obvious e part of the cycle in the area surrounded by the new cycle missing exactty this vertex. T! tis shows

csEnjectured that every hypohamiltonian graph has s conjecture was disprolyed in [ 141. We can now find a tlows: Wr, dt;sp two new edges in the unon (of HO such that this region is partitioned illto two SI region. Again by the Ctinberg criterion iltonian. Hence it is hypohamiltonian, [ II Ii ]I

o obtain hypohamiltonian graphs fram phs by replacing certain vertices by graphs of t:he Petersen g tph. It is easy to see that the Section 2 j remain valid if we instead of a vettexn graph use a graph obtained from any n graph by deleting any vertex of degree 3. Therefore, s labelled 3, 5, 17, 19, 24,26 of the graph 41 by a vertex-deleted su aph of HO, it fol’lows from the that wle get ZIplanar hy miltonian graph of girth 3. hypohamiltonian graph GO contains two , then by successively using the cunsttucrated in [ 13, Figs. 1, 21 ane obtains an n (respectively hyptraceable) graphs. atso the resulting hyphamiltonian and phs ate phar. So we have the following result:

( 1) Every hypohamiftonian graph is 3-connected. (2) If the deletion of three vertices X, y, z from a hypohamiltonian gra;ph leaves a disconnected graph, then no two of the vcrkes X, ,v, z are adjacent. (3) INa vertex of degree 3 in a hypohamiltonian graph is contained in a triangle. (4) If two adjacent vertices are deleted from a hypohamiltonian graph, then the resulting graph is 2-connected, Nate that (3) and (4) are immediai e consequences of (2). We now consider five mutuaily disioint hypohamiltonian graphs Gi, I < i G 5, such that Gi contains two adjacent vertices+ yi each of degree 3. Let l~l, iii (resp. of, di) detlote the neighbours of xi (KSP. yi) other than yj (WSY. Xi).BY (3), +, 6,, Ciedi a,redistinct. Let Hi denote q - {xi, y&. Sin : Gj k non-hamiltonian. we have: (5) H’ has no hamiltonian path connecting one of the vertices +, bi with one of the vertices cl,dia (6) If Hi has a spanning subgraph consisting of two disjoint paths each of which connects two vertices of ai, b, ci, di, then one of the paths connectsq a.nd bj and the other connects ci and di. Since each vertex-deleted subgrapk of Gi is hamiltonian, we have: (7) Hi has a hamiltonian path connecting ai with bi and a hamiltonian path connecting Ci with & (8) If z is any vertex of Hi, then either I;ri - z has a hamiltonian path mnnecting one of ai,, bi with one of t:*i, di or else Ip;:- z has a spanning subgraph consisting of two disjoint paths each connecting one of aj, b, with one of cl, dj* NOW furm U;=, Hi and add the edl es (Cj, Qi+1)stdi, bi+$ for 1 G i G 5 (the indices being expressed module 5). The resulting graph is denoted by T. If, in particular, each Gj is isomorphic to the Petersen graph then T is isomorphic to the hypotraceable graph of Worton 19, Fig. I]* (41, each Hi is Z-connected, so T is Zkzonne\=ted. Furthermore, if e Gj is cubic then T is cubic, anti if eac;h G, is lanar then T can be: to be planar.

e shall first prove that T has no hamiltonian

s joining H3

u H,$ ta Hz w H5. It is easy to see “chat the

P csntains exactly two of these c ges csntmdicts either lean assume that contains nP1four ed s. Furthermore, (resp. P n H,) is a hamilwith b3 (resp. c4 with d4). must have une endvertex in W2 and the ut then P n H, is a hmiltoqiarm path of M, connectin clt, which contradicts (5). Henr=e T has ~l”to ham& that every vertex-deleted subgraph of 7 has a hamilt z be any vertex of 7; say z 6’ HI. Suppse first that ~amilto~ja~ path conne&ing or;e of the vertices q, 21, with It is an irntzediate consequence of (7) that e vertices q,

iarr cycle. in partke\l.ar, each of these graphs has a hamilh starting at any vertex. lt is now sasy to find a hamittonian - z hbs no hamiltonian path of the z has a spring subgraph conisjoint paths eacfn connecting one of the vertices ai, bl one of the vertices cl, d, . Then by (7),

rail tanian cycle. Also,

ny csnniected graph which has a s two &&int cyicles has a h~~?to~i~ pa roof is complete,

wajr is the graph of Horton which has 40 vertices. If each Gi is isornorphic to the graph MOof Fig. 1, then qweget a planar 3-Connected hypotraceable graph with 5 15 vertices.

Let G (respectively G’) be a graph containing a vertex u (respectively u’j jziined to vertices X, y, z (respectively A’, y’, z’) and to no other vertex, Form a new aph C” from G - u and 6’ - u’ by identifyingx and intc3 a vertex x y and y’ into a vertex y” and t aisd z’ into a vertex 2”. e shall then say that C” is obtained from G by replacing u by a vertexdeleted subgraph of G’. The special case where G’ is the Petersen graph is lillustrated ira [ 14, Fig. t 1. The following result can be regarded as an infinite version of [ 14, Theorem 1). /

ProoR Suppose first that G” has a 2-way infinite hamiltonian path P. For each $4 E A, let pU denote the subpatk of P which in G, connects two vertices adjacent (in G,) to u,. Since C, is not hamiltonisn. !$ does not

) we obtain a 2-w(ay inf’?nik

iTi&. 2. Aa infhitc

hypohmiHortian graph.

~~~~~cting .Xand y and Gontaining all vertices of Gu - z.+~ except Pu. Then we place for each u E the segment xtay of P a>>* -Way itrfinite hamil niarr path of C’ - 2. Suppose next u,) -- V(G) for some u F A. Let X, y, IVdenote the vertices zi, has in common with G - A. Since G, is hypcr:ramiltol\ian, al;,,w-Vu - z contains a hamiltonian path Pu connecting two of the vertices r -way infinite hamilton:ian path xandy).G-ruts vlr’for each u’ E A - {u} we define the rgumen, and by a similar modification Main a infinite hamiltonkn path of G’ - z. ’ is hypohamittotiim.

btai.ned from a 2-way infinite path ew vertex and joining it to each :ci for i clearly (i), (ii) and (iii) of Theorem 4.1 vertex of A by a vert -deleted subgraph .1 an infinite gh we obtain by The Since there is more than one non-planar and hami~t~n~a~ graph the corollary follows. ing each vertex of C is shown in Fig. 2.

e shall first characterize the 2-fragments of finite hypotraceable graphs. An example of a finite hypotraceable aph of connectivity 2 is shown in [ 13, Fig. 3).’

Pkaof. Suppose first that G is hypotraceable. Since G - y has a hamiltonian path, G, - y has a hamiltoni starting at x ! w i = I, 2. Paths Pi (r’==1, 2) are defined anal G, has a hamifarbnian path P starting at x, then P WPz woul iltonian path of G, a contradiction. By symmetry (i) follows Now consider any vertex z E say z E V(C, ). G - z has a hamillonian path R e claim that P CI G, is a hamiltonian path of G, - z stat at x or y. ff not, th sists of two disjoint paths (one of‘ ch might have leng P R C, is then a hamiltonian path of G, starting at x or this contradicts (i), and (ii) is prov_;:d. For the second part of the lemma suppose (i) and (ii) c’re satisfie shaI1 first prove by reductio ad absurdurn G has no hamiltonia Suppose P is a hamiltonian path of 48.Th contain; a subpath connectingx and y. Supp ,se withour 10s~of rality that this path is contained in G,. r~ C, is ;aLaw:?anian path of G2 or y, a contradiction which proves tha.t G has no hamilton let z be any vertex of G, say z E IF(&). G1 starting at x or y (say at x), and G2 - y has starting at x. Then P, u P2 is a hamiltsnian is proved. he next lemma c

rst that G is hypotraceable. Since C - - sf has a I-way inmiltorGan path, one of G,, G, is fin& and the other is infinite. se G, is finite. Now (i) and (ii) can be proved aimost word for Iso th.e second par of Lemma 5.3 is Iproved as ra of Lemma 5. I * ditions (i), (ii) in f? erk$z !Xtwlzen x and pF.

emmas zW,5.2,5.3

imply that

raceable graph and a 2-frag-

Fig. 3. An infinite hypotraceable graph.

traceable graph obtained in this way from the finite af [ 13, Fig. 31 and the infinite hypohamiltonian graph of Fig. 2 is shown in Eig. 3. If we take two finite 2-fragments of illfinite hypotraceable graphs and identify their respective vertices of attachment, then by the first part of Lemma 5.3 and the skcond part of Lemma 5.1 the resulting graph is ypotraccabte. Since the infinite hypotraceable graphs we nitePy matliy finite 2-fragments, we obtain in this ily of finite otraceable graphs. Using the graph of Fig. 3 we can instance obtain the hypotraceable graph shown in Fig, 4. This aph has 41 vertices. Theorem 3.2 in [ 131 asserts that there exist a. finite hypotr ceable graph with p vertkes for p = 34,37,39,40 and fur eachp 3 42. Thus we have obtained a slight improvement of this result.

do

of hy~~~ami~t~nian and hy $$ree3 have played an important role ph contains a vz-fex of degree 3. Does every finite h;ypohamil$onian graph contain a vertex

ve been able to answer thl: question in the affirmative for planar

oes there exist a finite planar, cubic hypoham’ltonian ively hypotraceabie) graph (compare ChvSitalt[ 2]), finite hy~h~~~ni~t~nian graph is clearly 3-connected and cycli-

anrd S-connected cubic,, planar, nonhamiltonian

2 (mod 3), whrile the distinguished region is bounded by r ‘ter 3 2 (mod 3). Consider f”& the case r = 0 (mod 3) If we afe a new region such that the number of edges bounding 3). Hence the resultin ph is nanhamiltonian by the ich is not adjacent to the distinguished region but joined by tex qwh.kh is [email protected]~* to this region, then we create a new

We conclude with a problem on inf’Ste hypohamiltonian tively hypotrxxable graphs). .A Dues there exist an infinite, locally finite hypohamiltonian (respectively hypotracenble) graph.

f 1f V. chv&ak, Flip-flops in kyphamihnian

graphs,Can. Math. Buil. 16 (197 9) 33-4 1. hamiltonian graph theory, in: F. Harary (ed.). New Directions in Graph Theory (Aademic Press,New York, 1973,. (31 V, Chv&af, D.A. Warnerand D.E. Knuth, Selected combinatoriat research problems, Stanford University, Stanford (X972), Problem 19. 141 J. Doyen aftd V. Van Diea, New famaies of hypohamiltanian graphs, Discrete Math* 13 (1975) 225-236. 1st C.B+ Faulkn~ and D,H. Ytounger, Non-has&o&n cubic planar maps, Disclcte Math. 7 (19743 67-34. [6] El &inberg, Pdane homogeneous graphsof degree three without Hamiltonhrl circuits, Latvian Math. Yewbook, Izdal. “Zinatne”, Riga 4 (1968) 51-58 (in Russiao~). f 7 I B, Griinb;tum,Polytopes, graphs and complexe%.Bulf. Am. Math. Sot. 76 ( 1970) 113 I- 120 1. [8j B. Griinbaum, Vertices missed by longest patkvor tlrcuits, J. Combin. Theory 17 (1974) 31.-38. 191 R. Guy, Monthly re%archproblems, 19651973, Am. Math. Monthly 80 (1973) 1120-112% ( 10) F. Narary, Graph Theory (Addison-Wesley, Reading, Mass., 1969). f 11 f J.C. Mets, J.P. Duby and F. Vigui, Recherchesystemntique des grapheshypohamikonian, in: P. Rosenstiehl fed.), Theox-it! de!: Graphes (Duaod, Mis, 1967) 153- 160. ansberger, &tJlcmatical Gems C th. Assw. sf Anwica, 1973) Chapter 7. [ 13 j CT Thvma.=n, Hypohamiltonian and hyptraceabfe , Discrete Math. 9 (1974) 91-96. 10 (1974) 383-390. [ 14 j C+ ThomaSen* On hypohamiitonian graphs, [a) V. Chvbtal, Mew dinxtims

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