Jackson, Bill

Whitehead, Carol A.

Numérisation de Documents Anciens Mathématiques

Some remarks on Jaeger's dual-hamiltonian conjecture

Annales de l’institut Fourier (AIF)

ANNALES DE L’INSTITUT FOURIERBILL JACKSONCAROL A. WHITEHEADSome remarks on Jaeger’s dual-hamiltonianconjectureAnnales de l’institut Fourier, tome 49, no 3 (1999), p. 921-926<http://www.numdam.org/item?id=AIF_1999__49_3_921_0>© Annales de l’institut Fourier, 1999, tous droits réservés.L’accès aux archives de la revue « Annales de l’institut Fourier »(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-tion commerciale ou impression systématique est constitutive d’une in-fraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Ann. Inst. Fourier, Grenoble49, 3 (1999), 921-926SOME REMARKS ON JAEGERSDUAL-HAMILTONIAN CONJECTUREby Bill JACKSON and C.A. WHITEHEADWe consider finite simple graphs without loops or multiple edges. LetG be a connected graph. A cocircuit of G is a minimal set of edges of Gwhose removal disconnects G. Thus X is a cocircuit of G if and only ifG — X has exactly two components and every edge of X is incident withboth components. It follows that if X is a cocircuit then \X\ < \E\ — \V\-\-2^with equality if and only if both components of G — X are trees. We shallsay that X is a Hamilton cocircuit of G if \X\ = \E\ - \V\ +2 and, followingJaeger [3], that G is dual hamiltonian if G has a Hamilton cocircuit. Thesedefinitions are motivated by matroid duality, from which it follows that aplane graph is hamiltonian if and only if its dual graph is dual hamiltonian.Thus conditions implying hamiltonicity in plane graphs can be readilytranslated to conditions implying dual hamiltonicity in plane graphs. Wewill be concerned with a conjecture of Jaeger [3] which suggests that onesuch condition for dual hamiltonicity generalises to non-planar graphs. Tostate his conjecture we need one further definition. Suppose that G containstwo edge disjoint circuits. Then G is said to be cyclically k-connected ifwhenever we partition G into two subgraphs H^ and H^ both of whichcontain circuits, we have \V(H-i) H V{H'z)\ > k. Thus, if G is cyclically k-connected, then each circuit of G must have length at least k. The concept ofcyclic connectivity is the matroid dual to the 'standard5 definition of graphconnectivity, see [4]. In particular, a plane graph is /^-connected if and onlyif its dual is cyclically fc-connected. It follows that the classical result ofKeywords: Cubic graphs - Hamilton circuits - Cocircuits.Math. classification: 05C05 - 05C45.922 BILL JACKSON & C.A. WHITEHEADWhitney [7] that every 4-connected plane triangulation is hamiltonian isequivalent to:THEOREM 1. — Every cyclically 4-connected planar cubic graph isdual hamiltonian.The above mentioned conjecture of Jaeger is that this result remainsvalid for non-planar graphs.CONJECTURE 1. — Every cyclically 4-connected cubic graph is dualhamiltonian.Using Tutte's generalization [6] of Whitney's Theorem, we maydeduce that every cyclically 4-connected planar graph is dual hamiltonian.It is not true, however, that this more general result remains valid for non-planar graphs. To see this, consider the graph G whose vertex set is theunion of five disjoint independent sets of size three, 61, 62, . . . , 65, in whicheach vertex of Si is adjacent to every vertex of 6^4.1 for 1 <, i <, 5, wheresubscripts are to be read modulo five. Then G is cyclically 4-connected sinceG is 4-connected and has girth 4. Suppose G is dual hamiltonian. ThenV(G) can be partitioned into two subsets Ti.Fa, each of which inducesa tree in G. Without loss of generality we may assume |Ti D 611 ^> 2.Then \T^ H 62] > 2, otherwise Ti H (61 U 52) will induce a 4-circuit in G.Continuing this reasoning we eventually deduce that |Ti H 65] > 2. Butthen TI H (61 U 65) will induce a 4-circuit in G.For the remainder of this note, we shall restrict our attention tocubic graphs. For such graphs the concept of cyclic 4-connectivity can besimplified as follows: a cubic graph G -^ K^ is cyclically 4-connected if andonly if the only edge cuts in G with fewer than four edges are obtainedby taking three edges incident with some vertex. Jaeger's main concernin [3] involved the parameter 5(G), which he defined to be the size of alargest induced forest in G. He showed that, if G is a cubic graph, thens(G) <, |_(3|y| - 2)/4J, and furthermore that if G is dual hamiltonian thenequality holds. He was led to make Conjecture 1 by the result of Payan andSakarovitch [5] that s(G) = [{3\V\ - 2)/4J for all cyclically 4-connectedcubic graphs G. (There are many conjectures which would imply that thefamily of cyclically 4-connected cubic graphs has nicer properties than theentire family of 3-connected cubic graphs. The above mentioned result ofPayan and Sakarovitch is the only solid evidence we know of in support ofthis statement.) In the remainder of this note we shall make several remarksJAEGER'S DUAL-HAMILTONIAN CONJECTURE 923on Conjecture 1.The following result shows that Conjecture 1 is equivalent to aconjecture concerning an edge-partition of a cubic graph.LEMMA 2. — A cubic graph G is dual hamiltonian if and only if theedge set of G can be partitioned into two trees.Proof. — Suppose G is dual hamiltonian. Then V(G) can be parti-tioned into two induced trees T^,T^. Let H be the spanning subgraph ofG containing all edges which join T\ and T^. Then H is a bipartite graphof maximum degree two and so has a two edge colouring E\, E^. ThenE(T^) U J^i, E(T^ U £'2 is the required partition of E(G) into two trees.Suppose, on the other hand, that E(G) has a partition T[^ T^ into twotrees. Let Ti be the tree obtained from T[ by deleting all its end vertices.Then Ti, T^ is the required partition of V(G) into two induced trees. DIt follows that Conjecture 1 is equivalent to:CONJECTURE 2. — The edge set of every cyclically 4-connectedcubic graph G can be partitioned into two trees.Given a cubic graph G and e € E(G), let Ge be the cubic (multi)-graph obtained from G — e by suppressing its two vertices of degree two.One may try to use the following lemma as a basis for an inductive proofof Conjecture 1.LEMMA 3. — Let G be a cubic graph and e = v^v^ be an edge ofG. Let Ci be the edge of Ge 'containing7 Vi for 1 < i < 2. Suppose Gehas a Hamilton cocircuit containing e\. Then G has a Hamilton cocircuitcontaining e.Proof. — Let T\,T^ be a partition of V{Ge) into two induced treessuch that e\ is incident with both T\ and T^. We may suppose without lossof generality that either 62 is also incident with both T\ and T^ or that 62is an edge of the tree induced by T-z. In either case T\ + z^i, T^ + V2 is therequired partition of V(G). DAn obvious approach to using this lemma would be to try to use itto prove:924 BILL JACKSON & C.A. WHITEHEADCONJECTURE 3. — Let G be a cyclically 4-connected cubic graphand e e E(G). Then G has a Hamilton cocircuit containing e.This approach fails if Ge is not cyclically 4-connected. We have triedto get around this problem by working with various more complicatedinduction hypotheses, in line with either Conjecture 1 or 2. One suchattempt is described below.The problem with using Lemma 3 occurs when Ge no longer belongsto the family of graphs for which the inductive hypothesis applies. To getround this we could try to work with a larger family than the family ofcyclically 4-connected cubic graphs. To define such a family we proceed asfollows.Let G be a 3-connected cubic graph. Suppose that G is not cyclically4-connected and is not isomorphic to K^. Let X be a 'non-trivial' 3-edgecut in G and H\, H^ the two components of G — X. For 1 < i <^ 2 let Gibe the 3-connected cubic graph obtained from G by contracting H^-i to anew vertex v. We shall refer to v as a marker vertex in G\ and G^. We nowiterate this procedure for both G\ and G^. We continue until we obtain acollection of cubic graphs S each of which is either cyclically 4-connectedor else is isomorphic to K^. We shall refer to these graphs as pieces of G.Note that each 'non-trivial' 3-edge cut of G will be represented by a markervertex in exactly two pieces of G. We define a new graph D whose verticesare the pieces of (7, and in which two pieces are joined by an edge if theyhave a marker vertex in common. It follows from the decomposition theorydeveloped by Cunningham and Edmonds in [1] that D is a tree and that theset of pieces 5' and the tree D are uniquely defined by G. We shall refer toD as the decomposition tree of G and the pieces of G which correspond toend-vertices of D as end-pieces of G. We believe that if G is a 3-connectedcubic graph whose decomposition tree has maximum degree at most threethen G is dual hamiltonian. The bound on the maximum degree of thedecomposition tree comes from the construction described below.One may construct a 3-connected cubic graph H which is not dualhamiltonian by taking any 3-connected cubic graph G on at least sixvertices and 'blowing up5 each vertex v of G into a triangle Cy. To seethis we use Lemma 2. Suppose E(H) can be partitioned into two treesTi, T2. Since each triangle Cy in H must contain two edges of one tree andone edge of the other tree we see that Pi = Ti H E{G), P^ = T^ H E{G) isa partition of E(G) into two connected spanning subgraphs of maximumJAEGER'S DUAL-HAMILTONIAN CONJECTURE 925degree two. This is clearly impossible since G is cubic and has more thanfour vertices. If we begin with the triangular prism Go and apply thisconstruction recursively we obtain an infinite sequence of 3-connected cubicgraphs Go, Gi , . . . such that, for i > 1, Gi is not dual hamiltonian and hasa decomposition tree of maximum degree four. (The decomposition tree forGo is K^. The decomposition tree for G^+i can be obtained from that ofGi by attaching three new leaves to each of its end-vertices.)One can try to apply Lemma 3 inductively to the family of 3-connected cubic graphs whose decomposition tree has maximum degreethree by formulating a conjecture as follows:CONJECTURE 4. — Let G be 3-connected cubic graph whose de-composition tree D has maximum degree at most three and e be an edge ofG which lies in an end piece of G. Then G has a Hamilton cocircuit whichcontains e.Adopting the notation of Lemma 3, it can be seen that Ge will be a3-connected cubic graph with a decomposition tree D of maximum degreeat most three. Furthermore, either e\ or 63 will belong to an end piece ofGe unless the end piece Gi of G which contains e is isomorphic to K^ (thatis to say e belongs to a triangle of G), and the unique neighbour of Gi inD has degree three. Unfortunately, if this second alternative occurs, thenboth 61 and 63 will belong to a piece of Ge which has degree two in thedecomposition tree of Ge and the inductive argument again fails.Working with Yu [2], the first named author has recently verifiedConjecture 4 for planar graphs by proving the dual statement to:THEOREM 4. — Let G be a 3-connected planar cubic graph. Sup-pose that G has a decomposition tree D of maximum degree at most three.Let Gi be a piece of G corresponding to a vertex of D of degree at mosttwo, and e, f be edges both incident with a vertex of Gi which is not amarker vertex. Then G has a Hamilton cocircuit through e and f.Although this result gives some evidence in favour of Conjecture 4, itscurrent proof is discouraging since it makes even greater use of planaritythan Whitney's original theorem.926 BILL JACKSON & C.A. WHITEHEADBIBLIOGRAPHY[1] W.H. CUNNINGHAM and J. EDMONDS, A combinatorial decomposition theory,Canadian J. Math., 32 (1980), 734-765.[2] B. JACKSON and X. Yu, Hamilton cycles in plane triangulations, submitted.[3] F. JAEGER, On vertex induced forests in cubic graphs, Proc. Fifth SoutheasternConf. on Combinatorics, Graph Theory and Computing, Utilitas Mathematica,Winnipeg (1974), 501-512.[4] J.G. OXLEY, Matroid Theory, Oxford Univ. Press, Oxford, 1992.[5] C. PAYAN and M. SAKAROVITCH, Ensembles cycliquement stables et graphescubiques, Cahiers du C.E.R.O., 17 (1975), 319-343.[6] W. T. TUTTE, A theorem on planar graphs, Trans. Amer. Math. Soc., 82 (1956),99-116.[7] H. WHITNEY, A theorem on graphs, Ann. of Math., 32 (1931), 378-390.Bill JACKSON & C.A. WHITEHEAD,Goldsmiths CollegeDepartment of Mathematical andComputing SciencesLondon SE14 6NW (England).

https://aif.centre-mersenne.org/item/10.5802/aif.1699.pdf

Some remarks on Jaeger's dual-hamiltonian conjecture

Abstract

Similar works

Full text

Available Versions

Numérisation de Documents Anciens Mathématiques

Annales de l’institut Fourier (AIF)