Every graph occurs as an induced subgraph of some hypohamiltonian graph

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1 Reeive: 27 Oter 2016 Revise: 31 Oter 2017 Aepte: 20 Nvemer 2017 DOI: /jgt ARTICLE Ever grph urs s n inue sugrph f sme hphmiltnin grph Crl T. Zmfiresu 1 Tur I. Zmfiresu 2,3,4 1 Deprtment f Applie Mthemtis, Cmputer Siene n Sttistis, Ghent Universit, Krijgsln S9, 9000 Ghent, Belgium 2 Fhereih Mthemtik, Universität Drtmun, Drtmun, Germn 3 Simin Stilw Institute f Mthemtis, Rumnin Aem, Buhrest, Rumni 4 Cllege f Mthemtis n Infrmtin Siene, Heei Nrml Universit, Shijihung, P.R. Chin Crrespnene Crl T. Zmfiresu, Deprtment f Applie Mthemtis, Cmputer Siene n Sttistis, Ghent Universit, Krijgsln S9, 9000 Ghent, Belgium. Emil: mfiresu@gmil.m Funing infrmtin Fns Wetenshppelijk Onerek Astrt We prve the titulr sttement. This settles prlem f Chvátl frm 1973 n enmpsses erlier results f Thmssen, wh shwe it fr K 3, n Cllier n Shmeihel, wh prve it fr iprtite grphs. We ls shw tht fr ever uterplnr grph there eists plnr hphmiltnin grph ntining it s n inue sugrph. KEYWORDS hphmiltnin, inue sugrph MSC C10, 05C45, 05C60 1 INTRODUCTION Cnsier nn-hmiltnin grph G. We ll G hphmiltnin if fr ever verte v in G, the grph G v is hmiltnin. In similr spirit, G is si t e lmst hphmiltnin if there eists verte w in G, whih we will ll eeptinl, suh tht G w is nn-hmiltnin, n fr n verte v w in G, the grph G v is hmiltnin. Fr n verview f results n hphmiltniit till 1993, see the surve Hltn n Sheehn [7]. Fr newer mteril tht ls inlues wrk n the reentl intrue lmst hphmiltnin grphs, we refer the reer t [4,5,8,15] n the referenes fun therein. In 1973, Chvátl [2] ske whether ever grph m ur s the inue sugrph f sme hphmiltnin grph. As Thmssen writes in [10], n imprtnt prtil nswer ws prvie Cllier n Shmeihel [3] wh prve tht ever iprtite grph is n inue sugrph f sme hphmiltnin grph. In [11], Thmssen nstruts n infinite fmil f plnr ui hphmiltnin grphs n shws tht ertin eges n e e t these grphs suh tht the resulting grphs re hphmiltnin, s well. He uses this t give simple prf f the frementine result f Cllier n Shmeihel. J Grph Ther. 2017;1 7. wilenlinelirr.m/jurnl/jgt 2017 Wile Periils, In. 1

2 2 ZAMFIRESCU AND ZAMFIRESCU Erlier, Thmssen [9] h prven tht hphmiltnin grphs f girth 3 n 4 eist, i.e. tht K 3 n C 4 n ur s inue sugrphs f hphmiltnin grphs see ls [3]. This refute the njeture f Her et l. [6] tht hphmiltnin grphs hve girth t lest 5. Thmssen emphsies in [10] tht even fr K 4 the nswer t Chvátl's prlem is unknwn. In this nte, we prve tht n grph n pper s n inue sugrph f sme hphmiltnin grph. 2 AUXILIARY RESULTS Cnsier plnr lmst hphmiltnin grph with ui eeptinl verte, fr emple the grph F f rer 36 (isvere Gegeeur n the first uthr [5], n inepenentl Wiener [13, 14]), with eeptinl verte v. Nte tht F is the smllest knwn plnr grph fit fr the nstrutin t me hwever, there might e smller grphs usle tht hve nt een fun et. In ft, there is smller plnr lmst hphmiltnin grph knwn (fun Wiener [14] n f rer 31), ut it es nt ntin ui eeptinl verte, whih is neee fr the meth t wrk. v FIGURE 1 The p F f F. The eeptinl verte is mrke v Tke tw isjint pies F,F f F,withv V (F ) n v V (F ) rrespning t v. Put N(v )={,, } n N(v )={,,}. Fr n illustrtin f F, see Figure 1. Tke F v, F v, ientif with, n the eges n. The neighrs,,, re hsen suh tht we tin the grph H epite in Figure 2. (The hlf-eges shwn in Figures 2 n 3 en in verties utsie H.) Alre Thmssen use suh nstrutin in [9]. In wht fllws, we see F v n F v s sugrphs f H. In prtiulr,,,, ente verties in H, s well. Suppse n ritrr ut fie grph W hs hmiltnin le Λ, n let W inlue the grph H suh tht mng the verties f H nl,,, re nnete eges with verties in W H. Lemm 1. Either H Λis (i) the unin f tw isjint pths, ne frm t n the ther frm t,r(ii)pthfrm t,r(iii)pthfrm t. Prf. There is n hmiltnin pth p in H etween n, sine p wul hve t use the ege, ieling hmiltnin pth etween n in F v r hmiltnin pth etween n in F v (epening n when p piks up ). Aing the pth v r v, respetivel, we get hmiltnin le in F n ntritin is tine. Nw ssume there is hmiltnin pth q in

3 ZAMFIRESCU AND ZAMFIRESCU 3 FIGURE 2 We epit igrmmtill the grph H shwn ve with the retngle shwn elw FIGURE 3 There re tw essentill ifferent ws t trverse the retngle : in the first situtin ( ) we trverse isnnetel, while in the ltter tw ( r ) we trverse ignll H etween n. Then there eists hmiltnin pth etween n in F r hmiltnin pth etween n in F. As ve, we re le t ntritin sine F is nn-hmiltnin. If H Λis the unin f tw isjint pths, ne frm t, the ther frm t, then ne f these pths must ntin. Assume w.l..g. the frmer t e tht pth. Cnsiering it in F n ing t it the pth v, we tin hmiltnin le in F, ntritin sine F is lmst hphmiltnin. In se (i) we s tht H is isnnetel trverse, while if se (ii) r (iii) urs H is lle ignll trverse. In ses (ii) n (iii) f Lemm 1, when H is ignll trverse, we s mre preisel tht H is n trverse, respetivel. Cnsier the grph G f Figure 4. There, A, B, C,..., Q re grphs ismrphi t H. The length f the le Γ=w equls the numer f pies f H use t nstrut G, i.e. {A, B, C,,Q} = V (Γ). Γ is inlue in n ritrr hmiltnin grph Z with the hmiltnin le Γ. Lemm 2. The grph G is nt hmiltnin. Prf. Suppse G hs hmiltnin le Λ. Oviusl, nt ll pies f H re isnnetel trverse. Suppse A is trverse. Then Λ quikl visits, s it must ntinue with the pth u. IfB is ls ignll trverse, then nlgusl Λ quikl visits, n Λ is nt hmiltnin. Hene, Lemm 1, B is isnnetel trverse. Thus Λ.

4 4 ZAMFIRESCU AND ZAMFIRESCU B A u v w Q C Z FIGURE 4 The grph G Neessril, Q is trverse. S, nlgusl, Λ. Cntinuing this resning, we see tht n ege f Z is in Λ. Inee, ll ther pies f H re isnnetel trverse, s Λ et. Thus w Λ, n ntritin is tine. 3 MAIN RESULTS A grph is uterplnr if it pssesses plnr emeing in whih ever verte elngs t the unune fe. Nte tht grph is uterplnr if n nl if it es nt ntin grph hmemrphi t K 4 r K 2,3, see [1]. We nw present ur min therem. Therem 1. Ever grph is ntine in sme hphmiltnin grph s n inue sugrph. Prf. Let G e n ritrr grph. In the reminer f this prf, ll nttin refers t ntins intrue in Setin 2. Chse Z t e Γ t whih G is e in suh w tht the finite sequene f its verties is ple t ever sen verte f Γ. (S, the length f Γ is 2 V (G).) B Lemm 2, G is nt hmiltnin. It remins t prvie hmiltnin le in G s, fr ever verte s in G. A hmiltnin le f G w is shwn in Figure 4. B hnging u int uw we get hmiltnin le in G. Due t the smmetries, it remins t shw tht G s is hmiltnin fr n s V (F ). Cnsier s V (F ). There is hmiltnin pth in F s jining t, r t, r t. In the sen se ( t ), we hnge the rute f Λ insie the sugrph spnne V (A) {,, u, w} s shwn in Figure 5 (), n in the thir se ( t ), we hnge the rute s epite in Figure 5 (). In the first se, if s is the entrl verte f A, we tin hmiltnin le f G s piture in Figure 6. Fr ther psitins f s V (F ), see Figure 7. Frm the ve prf, we immeitel tin the fllwing.

5 ZAMFIRESCU AND ZAMFIRESCU 5 s s () () FIGURE 5 The rute f Λ insie the sugrph spnne V (A) {,, u, w} A B u v w Q C Z FIGURE 6 A hmiltnin le f G s FIGURE 7 A hmiltnin le f G s Therem 2. If G is n uterplnr grph, then there eists plnr hphmiltnin grph ntining G s n inue sugrph. Therem 2 nnt e etene t inlue ll plnr grphs ue t n elegnt rgument f Thmssen, wh prves in [11] tht Whitne's Therem [12] whih sttes tht plnr tringultins withut seprting tringles re hmiltnin, plnr tringultin nnt e n inue sugrph f n pl-

6 6 ZAMFIRESCU AND ZAMFIRESCU nr hphmiltnin grph. Hwever, Therem 1, n plnr tringultin is ver well n inue sugrph f sme (neessril nnplnr) hphmiltnin grph. Thmssen's results frm [11] n e use t esrie ertin iprtite plnr grphs tht n e sugrphs f plnr hphmiltnin grphs. He ges n t write: Me this is the se fr ever iprtite plnr grph. This questin remins unreslve, s fr instne K 2,3 is plnr et neither plnr tringultin, nr mng Thmssen's frementine plnr iprtite grphs, nr n uterplnr grph. Therem 3. (i) There eists hphmiltnin grph f rer 20n ntining K n s n inue sugrph. In prtiulr, there eists hphmiltnin grph f rer 80 ntining K 4. (ii) There eists plnr hphmiltnin grph f girth 3 n rer 216. (iii) Fr n uterplnr grph G f rer n there eists plnr hphmiltnin grph f rer 144n ntining G s n inue sugrph. Prf. Fr (i), we m mif the nstrutin frm the prf f Therem 1 using ever verte f Γ (sine G is in this se mplete grph). Thus, here the length f Γ is V (K n ) = n. Furthermre, we use the Petersen grph inste f the grph frm Figure 1. It is nw es t verif tht the rer f G is inee 20n. The prf f (ii) is the sme s the prf f (i), ut we reple Petersen's grph the plne lmst hphmiltnin grph shwn in Figure 1. Here, n =3. In (iii), the length f Γ is 2n.Weuse2npies f the grph frm Figure 2, whih is f rer 69. We tin grph f rer ((69 + 4) 2n) 2n = 144n. Prt (ii) imprves un given in [4, Crllr 3.4]. We en this nte with the fllwing. Prlem. Chrterie thse plnr grphs tht ur s inue sugrphs f plnr hphmiltnin grphs. ACKNOWLEDGMENT The reserh f Crl T. Zmfiresu is supprte Psttrl Fellwship f the Reserh Funtin Flners (FWO). ORCID Crl T. Zmfiresu REFERENCES [1] G. Chrtrn n F. Hrr, Plnr permuttin grphs, Ann. Inst. Henri Pinré, Nuv. Sér. Set. B 3 (1967), [2] V. Chvátl, Flip-flps in hphmiltnin grphs, Cn. Mth. Bull. 16 (1973), [3] J. B. Cllier n E. F. Shmeihel, New flip-flp nstrutins fr hphmiltnin grphs, Disrete Mth. 18 (1977), [4] J. Gegeeur n C. T. Zmfiresu, Imprve uns fr hphmiltnin grphs, Ars Mth. Cntemp. 13 (2017), [5] J. Gegeeur n C. T. Zmfiresu, On lmst hphmiltnin grphs. Sumitte. See Setin 3 f rxiv: [mth.co].

7 ZAMFIRESCU AND ZAMFIRESCU 7 [6] J. C. Her, J. J. Du, n F. Vigué, Reherhe sstémtique es grphes hphmiltniens, in: Ther f Grphs: Interntinl Smpsium, Rme, 1966, pp , Grn n Breh, New Yrk, n Dun, Pris, [7] D. A. Hltn n J. Sheehn, The Petersen grph, Chpter 7: Hphmiltnin grphs, Cmrige Universit Press, New Yrk, [8] M. Jneh et l., Plnr hphmiltnin grphs n 40 verties,j.grphther84 (2017), [9] C. Thmssen, On hphmiltnin grphs, Disrete Mth. 10 (1974), [10] C. Thmssen, Hphmiltnin grphs n igrphs. Ther n Applitins f Grphs, Leture Ntes in Mthemtis, 642, Springer, Berlin (1978), pp [11] C. Thmssen, Plnr ui hphmiltnin n hptrele grphs, J. Cmin. Ther Ser. B 30 (1981), [12] H. Whitne, A therem n grphs, Ann. Mth. 32 (1931), [13] G. Wiener, On nstrutins f hptrele grphs, Eletrn. Ntes Disrete Mth. 54 (2016), [14] G. Wiener, New nstrutins f hphmiltnin n hptrele grphs, T pper in J. Grph Ther, [15] C. T. Zmfiresu, Hphmiltnin n lmst hphmiltnin grphs, J.GrphTher79 (2015), Hw t ite this rtile: Zmfiresu CT, Zmfiresu TI. Ever grph urs s n inue sugrph f sme hphmiltnin grph. J Grph Ther. 2017;00: /jgt.22228