We prove the titular statement. This settles a problem of Chvátal from 1973 and encompasses earlier results of Thomassen, who showed it for K_3, and Collier and Schmeichel, who proved it for bipartite graphs. We also show that for every outerplanar graph there exists a planar hypohamiltonian graph containing it as an induced subgraph

Zamfirescu, Carol

Zamfirescu, Tudor I

Ghent University Academic Bibliography

Received: 27 October 2016 Revised: 31 October 2017 Accepted: 20 November 2017DOI: 10.1002/jgt.22228ART ICLEEvery graph occurs as an induced subgraph of somehypohamiltonian graphCarol T. Zamﬁrescu1 Tudor I. Zamﬁrescu2,3,41Department of Applied Mathematics, Com-puter Science and Statistics, Ghent University,Krijgslaan 281 - S9, 9000 Ghent, Belgium2Fachbereich Mathematik, UniversitätDortmund, 44221 Dortmund, Germany3“Simion Stoilow” Institute of Mathematics,Roumanian Academy, Bucharest, Roumania4College of Mathematics and InformationScience, Hebei Normal University, 050024Shijiazhuang, P.R. ChinaCorrespondenceCarol T. Zamﬁrescu,Department ofAppliedMathematics, Computer Sci-ence andStatistics,GhentUniversity,Krijgslaan 281 - S9, 9000Ghent, Belgium.Email: czamﬁrescu@gmail.comFunding informationFondsWetenschappelijkOnderzoekAbstractWe prove the titular statement. This settles a problemof Chvátal from 1973 and encompasses earlier resultsof Thomassen, who showed it for 𝐾3, and Collier andSchmeichel, who proved it for bipartite graphs. We alsoshow that for every outerplanar graph there exists a pla-nar hypohamiltonian graph containing it as an inducedsubgraph.KEYWORD Shypohamiltonian, induced subgraphMSC 2 0 1 0 .05C10, 05C45, 05C601 INTRODUCTIONConsider a non-hamiltonian graph 𝐺. We call 𝐺 hypohamiltonian if for every vertex 𝑣 in 𝐺, the graph𝐺 − 𝑣 is hamiltonian. In similar spirit, 𝐺 is said to be almost hypohamiltonian if there exists a vertex𝑤 in 𝐺, which we will call exceptional, such that 𝐺 −𝑤 is non-hamiltonian, and for any vertex 𝑣 ≠ 𝑤in 𝐺, the graph 𝐺 − 𝑣 is hamiltonian. For an overview of results on hypohamiltonicity till 1993, seethe survey by Holton and Sheehan [7]. For newer material that also includes work on the recentlyintroduced almost hypohamiltonian graphs, we refer the reader to [4,5,8,15] and the references foundtherein.In 1973, Chvátal [2] asked whether every graph may occur as the induced subgraph of some hypo-hamiltonian graph. As Thomassen writes in [10], an important partial answer was provided by Collierand Schmeichel [3] who proved that every bipartite graph is an induced subgraph of some hypohamil-tonian graph. In [11], Thomassen constructs an inﬁnite family of planar cubic hypohamiltonian graphsand shows that certain edges can be added to these graphs such that the resulting graphs are hypo-hamiltonian, as well. He uses this to give a simple proof of the aforementioned result of Collier andSchmeichel.J Graph Theory. 2018;88:551–557. wileyonlinelibrary.com/journal/jgt © 2017 Wiley Periodicals, Inc. 551552 ZAMFIRESCU AND ZAMFIRESCUEarlier, Thomassen [9] had proven that hypohamiltonian graphs of girth 3 and 4 exist, i.e. that 𝐾3and 𝐶4 can occur as induced subgraphs of hypohamiltonian graphs—see also [3]. This refuted theconjecture of Herz et al. [6] that hypohamiltonian graphs have girth at least 5. Thomassen emphasizesin [10] that even for 𝐾4 the answer to Chvátal's problem is unknown.In this note, we prove that any graph can appear as an induced subgraph of some hypohamiltoniangraph.2 AUXILIARY RESULTSConsider a planar almost hypohamiltonian graph with a cubic exceptional vertex, for example the graph𝐹 of order 36 (discovered by Goedgebeur and the ﬁrst author [5], and independently by Wiener [13,14]), with exceptional vertex 𝑣. Note that 𝐹 is the smallest known planar graph ﬁt for the constructionto come—however, there might be smaller graphs usable that have not been found yet. In fact, there isa smaller planar almost hypohamiltonian graph known (found by Wiener [14] and of order 31), but itdoes not contain a cubic exceptional vertex, which is needed for the method to work.oadvF IGURE 1 The copy 𝐹 ′′ of 𝐹 . The exceptional vertex is marked 𝑣′′Take two disjoint copies 𝐹 ′, 𝐹 ′′ of 𝐹 , with 𝑣′ ∈ 𝑉 (𝐹 ′) and 𝑣′′ ∈ 𝑉 (𝐹 ′′) corresponding to 𝑣. Put𝑁(𝑣′) = {𝑜, 𝑏, 𝑐} and 𝑁(𝑣′′) = {𝑜′, 𝑑, 𝑎}. For an illustration of 𝐹 ′′, see Figure 1. Take 𝐹 ′ − 𝑣′, 𝐹 ′′ −𝑣′′, identify 𝑜 with 𝑜′, and add the edges 𝑎𝑏 and 𝑐𝑑. The neighbors 𝑎, 𝑏, 𝑐, 𝑑 are chosen such that weobtain the graph 𝐻 depicted in Figure 2. (The “half-edges” shown in Figures 2 and 3 end in verticesoutside 𝐻 .) Already Thomassen used such a construction in [9]. In what follows, we see 𝐹 ′ − 𝑣′ and𝐹 ′′ − 𝑣′′ as subgraphs of 𝐻 . In particular, 𝑎, 𝑏, 𝑐, 𝑑 denote vertices in 𝐻 , as well.Suppose an arbitrary but ﬁxed graph 𝑊 has a hamiltonian cycle Λ, and let 𝑊 include the graph 𝐻such that among the vertices of 𝐻 only 𝑎, 𝑏, 𝑐, 𝑑 are connected by edges with vertices in 𝑊 −𝐻 .Lemma 1. Either𝐻 ∩ Λ is (i) the union of two disjoint paths, one from 𝑎 to 𝑏 and the other from 𝑐 to𝑑, or (ii) a path from 𝑎 to 𝑐, or (iii) a path from 𝑏 to 𝑑.Proof. There is no hamiltonian path 𝔭 in 𝐻 between 𝑎 and 𝑏, since 𝔭 would have to use the edge𝑑𝑐, yielding a hamiltonian path between 𝑎 and 𝑑 in 𝐹 ′′ − 𝑣′′ or a hamiltonian path between 𝑐 and 𝑏in 𝐹 ′ − 𝑣′ (depending on when 𝔭 picks up 𝑜). Adding the path 𝑎𝑣′′𝑑 or 𝑏𝑣′𝑐, respectively, we get ahamiltonian cycle in 𝐹 and a contradiction is obtained. Now assume there is a hamiltonian path 𝔮 inZAMFIRESCU AND ZAMFIRESCU 553oa bcd↓abcdF IGURE 2 We depict diagrammatically the graph 𝐻 shown above with the rectangle shown belowabcdabcdabcdF IGURE 3 There are two essentially diﬀerent ways to traverse the rectangle 𝑎𝑏𝑐𝑑: in the ﬁrst situation (𝑎𝑏 ∪ 𝑐𝑑)we traverse disconnectedly, while in the latter two (𝑎𝑐 or 𝑏𝑑) we traverse diagonally𝐻 between 𝑎 and 𝑑. Then there exists a hamiltonian path between 𝑜 and 𝑏 in 𝐹 ′ or a hamiltonian pathbetween 𝑜 and 𝑐 in 𝐹 ′. As above, we are led to a contradiction since 𝐹 is non-hamiltonian.If 𝐻 ∩ Λ is the union of two disjoint paths, one from 𝑏 to 𝑐, the other from 𝑑 to 𝑎, then one ofthese paths must contain 𝑜. Assume w.l.o.g. the former to be that path. Considering it in 𝐹 ′ andadding to it the path 𝑏𝑣′𝑐, we obtain a hamiltonian cycle in 𝐹 ′, a contradiction since 𝐹 ′ is almosthypohamiltonian. ■In case (i) we say that 𝐻 is disconnectedly traversed, while if case (ii) or (iii) occurs 𝐻 is calleddiagonally traversed. In cases (ii) and (iii) of Lemma 1, when 𝐻 is diagonally traversed, we say moreprecisely that 𝐻 is 𝑎 − 𝑐 and 𝑏 − 𝑑 traversed, respectively.Consider the graph 𝐺∗ of Figure 4. There, 𝐴, 𝐵, 𝐶 , ..., 𝑄 are graphs isomorphic to 𝐻 . Thelength of the cycle Γ = 𝑤𝑦𝑦∗ … 𝑦′ equals the number of copies of 𝐻 used to construct 𝐺∗, i.e.|{𝐴,𝐵, 𝐶,… , 𝑄}| = |𝑉 (Γ)|. Γ is included in an arbitrary hamiltonian graph 𝑍 with the hamiltoniancycle Γ.Lemma 2. The graph 𝐺∗ is not hamiltonian.Proof. Suppose 𝐺∗ has a hamiltonian cycle Λ. Obviously, not all copies of 𝐻 are disconnectedlytraversed. Suppose 𝐴 is 𝑎 − 𝑐 traversed. Then Λ quickly visits 𝑦, as it must continue with the path𝑐𝑢𝑥𝑦. If 𝐵 is also diagonally traversed, then analogously Λ quickly visits 𝑦, and Λ is not hamiltonian.Hence, by Lemma 1, 𝐵 is disconnectedly traversed. Thus 𝑥𝑦𝑧 ⊂ Λ.554 ZAMFIRESCU AND ZAMFIRESCUZCBAQx∗y∗z∗xyzacuwvdbxy zF IGURE 4 The graph 𝐺∗Necessarily, 𝑄 is 𝑏 − 𝑑 traversed. So, analogously, 𝑥′𝑦′𝑧′ ⊂ Λ. Continuing this reasoning, we seethat no edge of 𝑍 is in Λ. Indeed, all other copies of 𝐻 are disconnectedly traversed, so 𝑥∗𝑦∗𝑧∗ ⊂ Λetc. Thus 𝑤 ∉ Λ, and a contradiction is obtained. ■3 MAIN RESULTSA graph is outerplanar if it possesses a planar embedding in which every vertex belongs to theunbounded face. Note that a graph is outerplanar if and only if it does not contain a graph homeo-morphic to 𝐾4 or 𝐾2,3, see [1]. We now present our main theorem.Theorem 1. Every graph is contained in some hypohamiltonian graph as an induced subgraph.Proof. Let 𝐺 be an arbitrary graph. In the remainder of this proof, all notation refers to notions intro-duced in Section 2. Choose 𝑍 to be Γ to which 𝐺 is added in such a way that the ﬁnite sequence of itsvertices is placed at every second vertex of Γ. (So, the length of Γ is 2|𝑉 (𝐺)|.) By Lemma 2, 𝐺∗ is nothamiltonian. It remains to provide a hamiltonian cycle in 𝐺∗ − 𝑠, for every vertex 𝑠 in 𝐺∗.A hamiltonian cycle of𝐺∗ −𝑤 is shown in Figure 4. By changing 𝑢𝑥𝑦 into 𝑢𝑤𝑦we get a hamiltoniancycle in 𝐺∗ − 𝑥.Due to the symmetries, it remains to show that 𝐺∗ − 𝑠 is hamiltonian for any 𝑠 ∈ 𝑉 (𝐹 ′). Consider𝑠 ∈ 𝑉 (𝐹 ′). There is a hamiltonian path in 𝐹 ′ − 𝑠 joining 𝑏 to 𝑐, or 𝑐 to 𝑜, or 𝑜 to 𝑏. In the second case(𝑐 to 𝑜), we change the route of Λ inside the subgraph spanned by 𝑉 (𝐴) ∪ {𝑥, 𝑦, 𝑢, 𝑤} as shown inFigure 5 (a), and in the third case (𝑜 to 𝑏), we change the route as depicted in Figure 5 (b). In the ﬁrstcase, if 𝑠 is the central vertex 𝑜 of 𝐴, we obtain a hamiltonian cycle of 𝐺∗ − 𝑜 as pictured in Figure 6.For other positions of 𝑠 ∈ 𝑉 (𝐹 ′), see Figure 7. ■From the above proof, we immediately obtain the following.ZAMFIRESCU AND ZAMFIRESCU 555abcdo sabcdos(a) (b)F IGURE 5 The route of Λ inside the subgraph spanned by 𝑉 (𝐴) ∪ {𝑥, 𝑦, 𝑢, 𝑤}ZCBAQx∗y∗z∗xyzacuwvdbxy zoF IGURE 6 A hamiltonian cycle of 𝐺∗ − 𝑜abcdo sF IGURE 7 A hamiltonian cycle of 𝐺∗ − 𝑠Theorem 2. If𝐺 is an outerplanar graph, then there exists a planar hypohamiltonian graph containing𝐺 as an induced subgraph.Theorem 2 cannot be extended to include all planar graphs due to an elegant argument of Thomassen,who proves in [11] that by Whitney's Theorem [12]—which states that planar triangulations withoutseparating triangles are hamiltonian—, a planar triangulation cannot be an induced subgraph of any pla-556 ZAMFIRESCU AND ZAMFIRESCUnar hypohamiltonian graph. However, by Theorem 1, any planar triangulation is very well an inducedsubgraph of some (necessarily nonplanar) hypohamiltonian graph.Thomassen's results from [11] can be used to describe certain bipartite planar graphs that can besubgraphs of planar hypohamiltonian graphs. He goes on to write: “Maybe this is the case for everybipartite planar graph.” This question remains unresolved, as for instance 𝐾2,3 is planar yet neither aplanar triangulation, nor among Thomassen's aforementioned planar bipartite graphs, nor an outerpla-nar graph.Theorem 3.(i) There exists a hypohamiltonian graph of order 20𝑛 containing 𝐾𝑛 as an induced subgraph. Inparticular, there exists a hypohamiltonian graph of order 80 containing 𝐾4.(ii) There exists a planar hypohamiltonian graph of girth 3 and order 216.(iii) For an outerplanar graph𝐺 of order 𝑛 there exists a planar hypohamiltonian graph of order 144𝑛containing 𝐺 as an induced subgraph.Proof. For (i), we may modify the construction from the proof of Theorem 1 by using every vertex ofΓ (since 𝐺 is in this case a complete graph). Thus, here the length of Γ is |𝑉 (𝐾𝑛)| = 𝑛. Furthermore,we use the Petersen graph instead of the graph from Figure 1. It is now easy to verify that the order of𝐺∗ is indeed 20𝑛.The proof of (ii) is the same as the proof of (i), but we replace Petersen's graph by the plane almosthypohamiltonian graph shown in Figure 1. Here, 𝑛 = 3.In (iii), the length of Γ is 2𝑛. We use 2𝑛 copies of the graph from Figure 2, which is of order 69. Weobtain a graph of order ((69 + 4) ⋅ 2𝑛) − 2𝑛 = 144𝑛. ■Part (ii) improves a bound given in [4, Corollary 3.4]. We end this note with the following.Problem. Characterize those planar graphs that occur as induced subgraphs of planar hypohamiltoniangraphs.ACKNOWLEDGMENTThe research of Carol T. Zamﬁrescu is supported by a Postdoctoral Fellowship of the Research Foun-dation Flanders (FWO).ORCIDCarol T. Zamfirescu http://orcid.org/0000-0002-9673-410XREFERENCES[1] G. Chartrand and F. Harary, Planar permutation graphs, Ann. Inst. Henri Poincaré, Nouv. Sér. Sect. B 3 (1967),433–438.[2] V. Chvátal, Flip-flops in hypohamiltonian graphs, Canad. Math. Bull. 16 (1973), 33–41.[3] J. B. Collier and E. F. Schmeichel, New flip-flop constructions for hypohamiltonian graphs, Discrete Math. 18(1977), 265–271.[4] J. Goedgebeur and C. T. Zamﬁrescu, Improved bounds for hypohamiltonian graphs, Ars Math. Contemp. 13 (2017),235–257.[5] J. Goedgebeur and C. T. Zamﬁrescu, On almost hypohamiltonian graphs. Submitted. See Section 3 ofarXiv:1606.06577 [math.CO].ZAMFIRESCU AND ZAMFIRESCU 557[6] J. C. Herz, J. J. Duby, and F. Vigué, Recherche systématique des graphes hypohamiltoniens, in: Theory of Graphs:International Symposium, Rome, 1966, pp. 153–159, Gordon and Breach, New York, and Dunod, Paris, 1967.[7] D. A. Holton and J. Sheehan, The Petersen graph, Chapter 7: Hypohamiltonian graphs, Cambridge UniversityPress, New York, 1993.[8] M. Jooyandeh et al., Planar hypohamiltonian graphs on 40 vertices, J. Graph Theory 84 (2017), 121–133.[9] C. Thomassen, On hypohamiltonian graphs, Discrete Math. 10 (1974), 383–390.[10] C. Thomassen, Hypohamiltonian graphs and digraphs. Theory and Applications of Graphs, Lecture Notes in Math-ematics, 642, Springer, Berlin (1978), pp. 557–571.[11] C. Thomassen, Planar cubic hypohamiltonian and hypotraceable graphs, J. Combin. Theory Ser. B 30 (1981),36–44.[12] H. Whitney, A theorem on graphs, Ann. Math. 32 (1931), 378–390.[13] G. Wiener, On constructions of hypotraceable graphs, Electron. Notes Discrete Math. 54 (2016), 127–132.[14] G. Wiener, New constructions of hypohamiltonian and hypotraceable graphs, To appear in J. Graph Theory,https://doi.org/10.1002/jgt.22173[15] C. T. Zamﬁrescu, Hypohamiltonian and almost hypohamiltonian graphs, J. Graph Theory 79 (2015), 63–81.How to cite this article: Zamﬁrescu CT, Zamﬁrescu TI. Every graph occurs as an inducedsubgraph of some hypohamiltonian graph. J Graph Theory. 2018;88:551–557. https://doi.org/10.1002/jgt.22228

Every graph occurs as an induced subgraph of some hypohamiltonian graph

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Every graph occurs as an induced subgraph of some hypohamiltonian graph

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