- Email: [email protected]
2-Factor hamiltonian graphs
Journal of Combinatorial Theory, Series B 87 (2003) 138–144
http://www.elsevier.com/locate/jctb
2-Factor hamiltonian graphs M. Funk,a,1 Bill Jackson,b,* D. Labbate,a,2 and J. Sheehanc a b
Dipartimento di Matematica, Universita` della Basilicata, C. da Macchia Romana, 85100 Potenza, Italy School of Mathematical Sciences, Queen Mary College, University of London, Mile End Road, London E1 4NS, UK c Department of Mathematical Sciences, King’s College, Old Aberdeen AB24 3UE, UK Received 3 August 2000
Abstract The Heawood graph and K3;3 have the property that all of their 2-factors are Hamilton circuits. We call such graphs 2-factor hamiltonian. We prove that if G is a k-regular bipartite 2factor hamiltonian graph then either G is a circuit or k ¼ 3: Furthermore, we construct an infinite family of cubic bipartite 2-factor hamiltonian graphs based on the Heawood graph and K3;3 and conjecture that these are the only such graphs. r 2002 Elsevier Science (USA). All rights reserved.
1. Introduction All graphs considered are finite and simple (without loops or multiple edges). We shall use the term multigraph when multiple edges are permitted. Several authors have considered the number of Hamilton circuits in k-regular graphs and there are interesting and beautiful results and conjectures in the literature. In particular, C.A.B. Smith (1940, cf. [14]) proved that each edge of a 3regular multigraph lies in an even number of Hamilton circuits. This result was extended to multigraphs in which each vertex has odd degree by Thomason [11]. A multigraph with exactly one Hamilton circuit is said to be uniquely hamiltonian. Thomason’s result implies that there are no regular uniquely hamiltonian *Corresponding author. E-mail addresses: [email protected] (M. Funk), [email protected] (B. Jackson), [email protected] (D. Labbate), [email protected] (J. Sheehan). 1 This research was supported by the Italian Ministry MURST. 2 This research was carried out within the activity of G.N.S.A.G.A. (project ‘‘Calcolo Simbolico’’) of the Italian C.N.R. and supported by the Italian Ministry MURST. 0095-8956/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. doi:10.1016/S0095-8956(02)00031-X
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multigraphs of odd degree. In 1975, Sheehan [10] conjectured that there are no uniquely hamiltonian k-regular graphs for all integers kX3: This conjecture has been verified by Thomassen for bipartite graphs [12], (under the weaker hypothesis that G has minimum degree 3), and for k-regular graphs when kX300 [13]. In this paper, we study k-regular hamiltonian graphs with the property that all their 2-factors are Hamilton circuits. We call these graphs 2-factor hamiltonian. We will prove that if G is a k-regular bipartite 2-factor hamiltonian graph then either G is a circuit or k ¼ 3: 2. Preliminaries Let G be a bipartite graph with bipartition ðX ; Y Þ such that jX j ¼ jY j; and A be its adjacency matrix. In general 0pjdetðAÞjpperðAÞ: We say that G is det-extremal if jdetðAÞj ¼ perðAÞ: Let X ¼ fx1 ; x2 ; y; xn g and Y ¼ fy1 ; y2 ; y; yn g be the bipartition of G: For F a 1-factor of G define the sign of F ; sgnðF Þ; to be the sign of the permutation of f1; 2; y; ng corresponding to F : (Thus G is det-extremal if and only if all 1-factors of G have the same sign.) The following elementary result is a special case of [6, Lemma 8.3.1]. Lemma 2.1. Let F1 ; F2 be 1-factors in a bipartite graph G and t be the number of circuits in F1 ,F2 of length congruent to zero modulo four. Then sgnðF1 Þ sgnðF2 Þ ¼ ð1Þt : Let G; G1 ; G2 be graphs such that G1 -G2 ¼ |: Let yAV ðG1 Þ and xAV ðG2 Þ such that dG1 ðyÞ ¼ 3 ¼ dG2 ðxÞ: Let x1 ; x2 ; x3 be the neighbours of y in G1 and y1 ; y2 ; y3 be the neighbours of x in G2 : If G ¼ ðG1 yÞ,ðG2 xÞ,fx1 y1 ; x2 y2 ; x3 y3 g; then we say that G is a star product of G1 and G2 and write G ¼ ðG1 ; yÞ * ðG2 ; xÞ: The Heawood graph H0 is the bipartite graph associated with the point/line incidence matrix of the Fano plane PGð2; 2Þ: Let H be the class of graphs obtained from the Heawood graph by repeated star products. These graphs were used by McCuaig [8] to characterise the 3-connected cubic det-extremal bipartite graphs: Theorem 2.2 (McCuaig [8]). A 3-connected cubic bipartite graph is det-extremal if and only if it belongs to H: Note: Bipartite graphs G with the more general property that some of the entries in the adjacency matrix A of G can be changed from 1 to 1 in such a way that the resulting matrix An satisfies perðAÞ ¼ detðAn Þ have been characterised in [5,7,9]. 3. 2-Factor hamiltonian bipartite graphs We are interested in determining which regular graphs can be 2-factor hamiltonian. Examples of such graphs are K4 ; K5 ; K3;3 ; the Heawood graph H0 ; and the cubic graph of girth five obtained from a 9-circuit by adding three vertices,
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each joined to three vertices of the 9-circuit. (The latter graph is the ‘Triplex graph’ of Robertson, Seymour and Thomas.) The following property is easy to prove. Proposition 3.1. If a bipartite graph G can be represented as a star product G ¼ ðG1 ; yÞ * ðG2 ; xÞ; then G is 2-factor hamiltonian if and only if G1 and G2 are 2-factor hamiltonian. Note that K4 * K4 shows that the star product of two non-bipartite 2-factor hamiltonian graphs is not necessarily 2-factor hamiltonian. Using Proposition 3.1, we can construct an infinite family of 2-factor hamiltonian cubic bipartite graphs by taking iterated star products of K3;3 and H0 : We conjecture that these are the only non-trivial 2-factor hamiltonian regular bipartite graphs. Conjecture 3.2. Let G be a 2-factor hamiltonian k-regular bipartite graph. Then either k ¼ 2 and G is a circuit or k ¼ 3 and G can be obtained from K3;3 and H0 by repeated star products. We shall show that there are no 2-factor hamiltonian k-regular bipartite graphs for kX4: We need the following elementary result. Lemma 3.3. Let G be a 2-factor hamiltonian cubic bipartite graph. Then G is 3connected and jV ðGÞj 2 ðmod 4Þ: Proof. Since G is hamiltonian, it is 2-connected. Suppose G is not 3-connected. Since G is cubic, it must have a 2-edge-cut fe; f g: Since G is regular and bipartite, e is contained in a 1-factor F of G: Then G F is a 2-factor of G which is not a Hamilton circuit. Thus G is 3-connected. Suppose jV ðGÞj 0 ðmod 4Þ: Let fF1 ; F2 ; F3 g be a 1-factorization of G: Then Fi ,Fj is a Hamilton circuit for all 1piojp3: By Lemma 2.1, sgnðFi ÞasgnðFj Þ for all 1piojp3: This is impossible since sgnðFi ÞAf1; 1g for all 1pip3: & We next show that graphs obtained by taking star products of H0 are ‘maximally’ 2-factor hamiltonian. This will follow from assertion ðaÞðiÞ in the following lemma. Unfortunately, our inductive proof of this statement seems to also require the other four assertions given in the lemma. Lemma 3.4. Let HAH; ðX ; Y Þ be the bipartition of H; and G be a multigraph obtained by adding an edge e with endvertices u; v to H: Let f ¼ uw be an edge of H adjacent to e in H þ e: (a) Suppose uAX and vAY : Then (i) G has a disconnected 2-factor containing e; (ii) G has a 2-factor containing e; f ;
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(iii) G has a factor F such that dF ðuÞ ¼ dF ðvÞ ¼ 4; dF ðzÞ ¼ 2 for all zau; v; and e; f are both contained in a common circuit of F : (b) Suppose u; vAX : Then (i) G has a factor F such that dF ðuÞ ¼ 4; dF ðzÞ ¼ 2 for all zau; and e; f are both contained in a common circuit of F ; (ii) G has a factor F such that dF ðvÞ ¼ 4; dF ðzÞ ¼ 2 for all zav; and e; f are both contained in a common circuit of F : Proof. We use induction on jV ðHÞj: If H ¼ H0 then it can be checked that the various assertions hold for H0 : (Using the 4-arc-transitivity of H0 ; see [15, p. 60], and the fact that H0 has diameter 3, it suffices to check the cases when u and v are joined by a path P in H0 of length 1, 2 or 3. In the first two cases there are two subcases depending on whether or not f AEðPÞ; in the third case we may assume by symmetry that f AEðPÞ:) Hence we may suppose that H has been constructed from H0 using star products. Choose a representation, H ¼ ðH1 ; yÞ * ðH2 ; xÞ; for H as a star product, where x1 ; x2 ; x3 AX are the neighbours of y in H1 ; y1 ; y2 ; y3 AY are the neighbours of x in H2 ; and x1 y1 ; x2 y2 ; x3 y3 AEðHÞ: Throughout the following proof we shall use the convention that edges denoted as pairs of vertices refer to edges in the ‘simple’ graphs H; H1 ; or H2 : Suppose e is only incident with vertices of H2 : We may apply induction to G2 ¼ H2 þ e to deduce that G2 has factors F2 satisfying each of the assertions in the lemma. Furthermore, since u; vAV ðHÞ; xefu; vg so dF2 ðxÞ ¼ 2: By symmetry we may assume that xy1 ; xy2 AEðF2 Þ: Let F1 be a 2-factor of H1 with yx1 ; yx2 AEðF1 Þ: Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; x2 y2 g is the required factor of G: Thus we may assume by symmetry that e is incident with exactly one vertex of H1 and exactly one vertex of H2 : Consider the following cases. Case 1: uAX -V ðH1 Þ and vAY -V ðH2 Þ: Let G1 be the multigraph obtained by adding an edge e1 with endvertices u; y to H1 ; and G2 be the multigraph obtained by adding an edge e2 with endvertices x; v to H2 : (a)(i) Applying (a)(i) inductively to G1 we obtain a disconnected 2-factor F1 in G1 such that e1 AEðF1 Þ: We may assume that yx1 AEðF1 Þ: Let f2 ¼ xy1 : Applying (a)(ii) inductively to G2 we obtain a 2-factor F2 in G2 such that e2 ; f2 AEðF2 Þ: Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; eg is the required disconnected 2-factor of G: In order to verify (a)(ii) and (a)(iii), we label an edge of G1 as f1 : If f1 AEðH1 Þ we put f1 ¼ f : Otherwise f ¼ xi yi for some iAf1; 2; 3g: Relabelling if necessary, we may assume that f ¼ x1 y1 ; and put f1 ¼ x1 y: (a)(ii) Applying (a)(ii) inductively to G1 we obtain a 2-factor F1 in G1 such that e1 ; f1 AEðF1 Þ: We now construct F2 and F as in (a)(i). (a)(iii) Applying (a)(iii) inductively to G1 we obtain F1 in G1 such that dF1 ðuÞ ¼ dF1 ðyÞ ¼ 4; dF1 ðzÞ ¼ 2 for all zau; y; and e1 ; f1 are both contained in a common circuit C of F1 : We may assume that yx1 AEðCÞ: Let f2 ¼ xy1 : We now proceed as in (a)(i), but applying (a)(iii) inductively to G2 rather than (a)(ii), and putting F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; x2 y2 ; x3 y3 ; eg:
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Case 2: uAX -V ðH2 Þ and vAY -V ðH1 Þ: Let G1 be the multigraph obtained by adding an edge e1 with endvertices v; y to H1 ; and G2 be the multigraph obtained by adding an edge e2 with endvertices x; u to H2 : Since uAX we have uayi for 1pip3: Thus f AEðH2 Þ and we put f2 ¼ f : (a)(i) Applying (b)(ii) inductively to G2 we obtain a factor F2 in G2 such that dF2 ðxÞ ¼ 4; dF2 ðzÞ ¼ 2 for all zax; and e2 ; f2 are both contained in a common circuit C of F2 : We may assume that xy1 AEðCÞ: Let f1 ¼ yx1 : Applying (b)(i) inductively to G1 (with the roles of X and Y reversed) we obtain a factor F1 in G1 such that dF1 ðyÞ ¼ 4; dF1 ðzÞ ¼ 2 for all zay; and e1 ; f1 are both contained in a common circuit of F1 : Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; x2 y2 ; x3 y3 ; eg is the required disconnected 2-factor of G: (a)(ii) The disconnected 2-factor F constructed in (a)(i) contains e; f : (a)(iii) Applying (b)(i) inductively to G2 we obtain a factor F2 in G2 such that dF2 ðuÞ ¼ 4; dF2 ðzÞ ¼ 2 for all zau; and e2 ; f2 are both contained in a common circuit C of F2 : We may assume that xy1 AEðCÞ: Let f1 ¼ yx1 : Applying (b)(ii) inductively to G1 we obtain a factor F1 in G1 such that dF1 ðvÞ ¼ 4; dF1 ðzÞ ¼ 2 for all zav; and e1 ; f1 are both contained in a common circuit of F1 : Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; eg is the required factor of G: Case 3: uAX -V ðH1 Þ and vAX -V ðH2 Þ: Let G1 be the multigraph obtained by adding an edge e1 with endvertices u; y to H1 ; and G2 be the multigraph obtained by adding an edge e2 with endvertices x; v to H2 : In order to verify (b), we label an edge of G1 as f1 : If f1 AEðH1 Þ we put f1 ¼ f : Otherwise f ¼ xi yi for some iAf1; 2; 3g: Relabelling if necessary, we may assume that f ¼ x1 y1 ; and put f1 ¼ x1 y: (b)(i) Applying (a)(iii) inductively to G1 we obtain a factor F1 in G1 such that dF1 ðuÞ ¼ dF1 ðyÞ ¼ 4; dF1 ðzÞ ¼ 2 for all zau; y; and e1 ; f1 are both contained in a common circuit C of F1 : We may assume that yx1 AEðCÞ: Let f2 ¼ xy1 : Applying (b)(i) inductively to G2 we obtain a factor F2 in G2 such that dF2 ðxÞ ¼ 4; dF2 ðzÞ ¼ 2 for all zax; and e2 ; f2 are both contained in a common circuit of F2 : Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; x2 y2 ; x3 y3 ; eg is the required factor of G: (b)(ii) Applying (a)(ii) inductively to G1 we obtain a 2-factor F1 in G1 such that e1 ; f1 AEðF1 Þ: (Note that e1 ; f1 are necessarily contained in a common circuit of F1 since F1 is a 2-factor and e1 ; f1 are both incident to u:) We may assume that yx1 AEðF1 Þ: Let f2 ¼ xy1 : Applying (b)(ii) inductively to G2 we obtain a factor F2 in G2 such that dF2 ðvÞ ¼ 4; dF2 ðzÞ ¼ 2 for all zav; and e2 ; f2 are both contained in a common circuit of F2 : Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; eg is the required factor of G: Case 4: uAX -V ðH2 Þ and vAX -V ðH1 Þ: Let G1 be the multigraph obtained by adding an edge e1 with endvertices v; y to H1 ; and G2 be the multigraph obtained by adding an edge e2 with endvertices x; u to H2 : Since uAX we have uayi for 1pip3: Thus f AEðH2 Þ and we put f2 ¼ f : (b)(i) We proceed in a similar fashion to Case 3 applying (b)(i) to G2 ; then (a)(ii) to G1 :
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(b)(ii) We proceed in a similar fashion to Case 3 applying (b)(ii) to G2 ; then (a)(iii) to G1 : & We may now prove our main result. Theorem 3.5. Let G be a 2-factor hamiltonian k-regular bipartite graph. Then kp3: Proof. Suppose that the theorem is false and let G be a counterexample chosen such that k is as small as possible. Let ðX ; Y Þ be the bipartition of G; where jX j ¼ jY j ¼ n: Let F be a 1-factor of G and H ¼ G F : Then H is a 2-factor hamiltonian ðk 1Þregular bipartite graph. Hence, by the minimality of k; we must have k ¼ 4: In addition, since H is cubic, it follows from Lemma 3.3, that jV ðHÞj 2 ðmod 4Þ and H is 3-connected. Let F 0 be a 1-factor of H: Since F ,F 0 is a Hamilton circuit of G and jV ðHÞj 2 ðmod 4Þ; we have sgnðF 0 Þ ¼ sgnðF Þ: Thus all 1-factors of H have the same sign and so H is det-extremal. Using Theorem 2.2 we deduce that HAH: We now obtain the required contradiction by applying Lemma 3.4(a)(i) to H; choosing e to be any edge of F : &
4. Minimally 1-factorable graphs Let G be a k-regular bipartite graph. We say that G is minimally 1-factorable if every 1-factor of G is contained in a unique 1-factorization of G: This paper was inspired by results on minimally 1-factorable graphs obtained in [1–4]. It can be seen that if G is minimally 1-factorable then G is 2-factor hamiltonian, and that, if k ¼ 2; 3; then G is minimally 1-factorable if and only if G is 2-factor hamiltonian. Using this observation, we could have deduced Lemma 3.3 from [3, Corollaries 2(iii) and 4]. Theorem 3.5 extends the result of [1] that minimally 1factorable k-regular bipartite graphs exist only when kp3: It follows from [2] that a smallest counterexample to Conjecture 3.2 is cubic and cyclically 4-edge connected, and from [4] that it has girth at least six. Thus, to prove the conjecture, it would suffice to show that the Heawood graph is the only 2-factor hamiltonian cyclically 4edge connected cubic bipartite graph of girth at least six.
References [1] M. Funk, D. Labbate, On minimally one-factorable r-regular bipartite graphs, Discrete Math. 216 (2000) 121–137. [2] D. Labbate, On 3-cut reductions of minimally 1-factorable cubic bigraphs, Discrete Math. 231 (2001) 303–310. [3] D. Labbate, On determinants and permanents of minimally 1-factorable cubic bipartite graphs, Note Math. 20 (2000/01) 37–42. [4] D. Labbate, Characterizing minimally 1-factorable r-regular bipartite graphs, Discrete Math. 248 (2002) 109–123. [5] C.H.C. Little, A characterization of convertible ð0; 1Þ-matrices, J. Combin. Theory Ser. B 18 (1975) 187–208.
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[6] L. Lova´sz, M.D. Plummer, Matching Theory, Annals of Discrete Mathematics, Vol. 29, NorthHolland, Amsterdam, 1986. [7] W. McCuaig, Po´lya’s permanent problem, submitted (1997). [8] W. McCuaig, Even dicycles, J. Graph Theory, to appear. [9] N. Robertson, P.D. Seymour, R. Thomas, Pfaffian orientations and even directed circuits, Ann. Math. 150 (1999) 929–975. [10] J. Sheehan, Problem section, in: C.St.J.A. Nash-Williams, J. Sheehan (Eds.), Proceedings of the Fifth British Combinatorial Conference, Congressus Numerantium XV, Utilitas Mathematica Publ. Corp., Winnipeg, 1975, p. 691. [11] A.G. Thomason, Hamiltonian cycles and uniquely 3-edge colourable graphs, in: B. Bolloba´s (Ed.), Advances in Graph Theory, Ann. Discrete Math. 3 (1978) 259–268. [12] C. Thomassen, On the number of hamiltonian cycles in cubic graphs, Combin. Probab. Comput. 5 (1996) 437–442. [13] C. Thomassen, Independent dominating sets and a second hamiltonian cycle, J. Combin. Theory Ser. B 72 (1998) 104–109. [14] W. T. Tutte, On hamiltonian circuits, J. London Math. Soc. 21 (1946) 98–101. [15] W.T. Tutte, Connectivity in Graphs, University of Toronto Press, Toronto, 1966.
http://www.elsevier.com/locate/jctb
2-Factor hamiltonian graphs M. Funk,a,1 Bill Jackson,b,* D. Labbate,a,2 and J. Sheehanc a b
Dipartimento di Matematica, Universita` della Basilicata, C. da Macchia Romana, 85100 Potenza, Italy School of Mathematical Sciences, Queen Mary College, University of London, Mile End Road, London E1 4NS, UK c Department of Mathematical Sciences, King’s College, Old Aberdeen AB24 3UE, UK Received 3 August 2000
Abstract The Heawood graph and K3;3 have the property that all of their 2-factors are Hamilton circuits. We call such graphs 2-factor hamiltonian. We prove that if G is a k-regular bipartite 2factor hamiltonian graph then either G is a circuit or k ¼ 3: Furthermore, we construct an infinite family of cubic bipartite 2-factor hamiltonian graphs based on the Heawood graph and K3;3 and conjecture that these are the only such graphs. r 2002 Elsevier Science (USA). All rights reserved.
1. Introduction All graphs considered are finite and simple (without loops or multiple edges). We shall use the term multigraph when multiple edges are permitted. Several authors have considered the number of Hamilton circuits in k-regular graphs and there are interesting and beautiful results and conjectures in the literature. In particular, C.A.B. Smith (1940, cf. [14]) proved that each edge of a 3regular multigraph lies in an even number of Hamilton circuits. This result was extended to multigraphs in which each vertex has odd degree by Thomason [11]. A multigraph with exactly one Hamilton circuit is said to be uniquely hamiltonian. Thomason’s result implies that there are no regular uniquely hamiltonian *Corresponding author. E-mail addresses: [email protected] (M. Funk), [email protected] (B. Jackson), [email protected] (D. Labbate), [email protected] (J. Sheehan). 1 This research was supported by the Italian Ministry MURST. 2 This research was carried out within the activity of G.N.S.A.G.A. (project ‘‘Calcolo Simbolico’’) of the Italian C.N.R. and supported by the Italian Ministry MURST. 0095-8956/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. doi:10.1016/S0095-8956(02)00031-X
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multigraphs of odd degree. In 1975, Sheehan [10] conjectured that there are no uniquely hamiltonian k-regular graphs for all integers kX3: This conjecture has been verified by Thomassen for bipartite graphs [12], (under the weaker hypothesis that G has minimum degree 3), and for k-regular graphs when kX300 [13]. In this paper, we study k-regular hamiltonian graphs with the property that all their 2-factors are Hamilton circuits. We call these graphs 2-factor hamiltonian. We will prove that if G is a k-regular bipartite 2-factor hamiltonian graph then either G is a circuit or k ¼ 3: 2. Preliminaries Let G be a bipartite graph with bipartition ðX ; Y Þ such that jX j ¼ jY j; and A be its adjacency matrix. In general 0pjdetðAÞjpperðAÞ: We say that G is det-extremal if jdetðAÞj ¼ perðAÞ: Let X ¼ fx1 ; x2 ; y; xn g and Y ¼ fy1 ; y2 ; y; yn g be the bipartition of G: For F a 1-factor of G define the sign of F ; sgnðF Þ; to be the sign of the permutation of f1; 2; y; ng corresponding to F : (Thus G is det-extremal if and only if all 1-factors of G have the same sign.) The following elementary result is a special case of [6, Lemma 8.3.1]. Lemma 2.1. Let F1 ; F2 be 1-factors in a bipartite graph G and t be the number of circuits in F1 ,F2 of length congruent to zero modulo four. Then sgnðF1 Þ sgnðF2 Þ ¼ ð1Þt : Let G; G1 ; G2 be graphs such that G1 -G2 ¼ |: Let yAV ðG1 Þ and xAV ðG2 Þ such that dG1 ðyÞ ¼ 3 ¼ dG2 ðxÞ: Let x1 ; x2 ; x3 be the neighbours of y in G1 and y1 ; y2 ; y3 be the neighbours of x in G2 : If G ¼ ðG1 yÞ,ðG2 xÞ,fx1 y1 ; x2 y2 ; x3 y3 g; then we say that G is a star product of G1 and G2 and write G ¼ ðG1 ; yÞ * ðG2 ; xÞ: The Heawood graph H0 is the bipartite graph associated with the point/line incidence matrix of the Fano plane PGð2; 2Þ: Let H be the class of graphs obtained from the Heawood graph by repeated star products. These graphs were used by McCuaig [8] to characterise the 3-connected cubic det-extremal bipartite graphs: Theorem 2.2 (McCuaig [8]). A 3-connected cubic bipartite graph is det-extremal if and only if it belongs to H: Note: Bipartite graphs G with the more general property that some of the entries in the adjacency matrix A of G can be changed from 1 to 1 in such a way that the resulting matrix An satisfies perðAÞ ¼ detðAn Þ have been characterised in [5,7,9]. 3. 2-Factor hamiltonian bipartite graphs We are interested in determining which regular graphs can be 2-factor hamiltonian. Examples of such graphs are K4 ; K5 ; K3;3 ; the Heawood graph H0 ; and the cubic graph of girth five obtained from a 9-circuit by adding three vertices,
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each joined to three vertices of the 9-circuit. (The latter graph is the ‘Triplex graph’ of Robertson, Seymour and Thomas.) The following property is easy to prove. Proposition 3.1. If a bipartite graph G can be represented as a star product G ¼ ðG1 ; yÞ * ðG2 ; xÞ; then G is 2-factor hamiltonian if and only if G1 and G2 are 2-factor hamiltonian. Note that K4 * K4 shows that the star product of two non-bipartite 2-factor hamiltonian graphs is not necessarily 2-factor hamiltonian. Using Proposition 3.1, we can construct an infinite family of 2-factor hamiltonian cubic bipartite graphs by taking iterated star products of K3;3 and H0 : We conjecture that these are the only non-trivial 2-factor hamiltonian regular bipartite graphs. Conjecture 3.2. Let G be a 2-factor hamiltonian k-regular bipartite graph. Then either k ¼ 2 and G is a circuit or k ¼ 3 and G can be obtained from K3;3 and H0 by repeated star products. We shall show that there are no 2-factor hamiltonian k-regular bipartite graphs for kX4: We need the following elementary result. Lemma 3.3. Let G be a 2-factor hamiltonian cubic bipartite graph. Then G is 3connected and jV ðGÞj 2 ðmod 4Þ: Proof. Since G is hamiltonian, it is 2-connected. Suppose G is not 3-connected. Since G is cubic, it must have a 2-edge-cut fe; f g: Since G is regular and bipartite, e is contained in a 1-factor F of G: Then G F is a 2-factor of G which is not a Hamilton circuit. Thus G is 3-connected. Suppose jV ðGÞj 0 ðmod 4Þ: Let fF1 ; F2 ; F3 g be a 1-factorization of G: Then Fi ,Fj is a Hamilton circuit for all 1piojp3: By Lemma 2.1, sgnðFi ÞasgnðFj Þ for all 1piojp3: This is impossible since sgnðFi ÞAf1; 1g for all 1pip3: & We next show that graphs obtained by taking star products of H0 are ‘maximally’ 2-factor hamiltonian. This will follow from assertion ðaÞðiÞ in the following lemma. Unfortunately, our inductive proof of this statement seems to also require the other four assertions given in the lemma. Lemma 3.4. Let HAH; ðX ; Y Þ be the bipartition of H; and G be a multigraph obtained by adding an edge e with endvertices u; v to H: Let f ¼ uw be an edge of H adjacent to e in H þ e: (a) Suppose uAX and vAY : Then (i) G has a disconnected 2-factor containing e; (ii) G has a 2-factor containing e; f ;
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(iii) G has a factor F such that dF ðuÞ ¼ dF ðvÞ ¼ 4; dF ðzÞ ¼ 2 for all zau; v; and e; f are both contained in a common circuit of F : (b) Suppose u; vAX : Then (i) G has a factor F such that dF ðuÞ ¼ 4; dF ðzÞ ¼ 2 for all zau; and e; f are both contained in a common circuit of F ; (ii) G has a factor F such that dF ðvÞ ¼ 4; dF ðzÞ ¼ 2 for all zav; and e; f are both contained in a common circuit of F : Proof. We use induction on jV ðHÞj: If H ¼ H0 then it can be checked that the various assertions hold for H0 : (Using the 4-arc-transitivity of H0 ; see [15, p. 60], and the fact that H0 has diameter 3, it suffices to check the cases when u and v are joined by a path P in H0 of length 1, 2 or 3. In the first two cases there are two subcases depending on whether or not f AEðPÞ; in the third case we may assume by symmetry that f AEðPÞ:) Hence we may suppose that H has been constructed from H0 using star products. Choose a representation, H ¼ ðH1 ; yÞ * ðH2 ; xÞ; for H as a star product, where x1 ; x2 ; x3 AX are the neighbours of y in H1 ; y1 ; y2 ; y3 AY are the neighbours of x in H2 ; and x1 y1 ; x2 y2 ; x3 y3 AEðHÞ: Throughout the following proof we shall use the convention that edges denoted as pairs of vertices refer to edges in the ‘simple’ graphs H; H1 ; or H2 : Suppose e is only incident with vertices of H2 : We may apply induction to G2 ¼ H2 þ e to deduce that G2 has factors F2 satisfying each of the assertions in the lemma. Furthermore, since u; vAV ðHÞ; xefu; vg so dF2 ðxÞ ¼ 2: By symmetry we may assume that xy1 ; xy2 AEðF2 Þ: Let F1 be a 2-factor of H1 with yx1 ; yx2 AEðF1 Þ: Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; x2 y2 g is the required factor of G: Thus we may assume by symmetry that e is incident with exactly one vertex of H1 and exactly one vertex of H2 : Consider the following cases. Case 1: uAX -V ðH1 Þ and vAY -V ðH2 Þ: Let G1 be the multigraph obtained by adding an edge e1 with endvertices u; y to H1 ; and G2 be the multigraph obtained by adding an edge e2 with endvertices x; v to H2 : (a)(i) Applying (a)(i) inductively to G1 we obtain a disconnected 2-factor F1 in G1 such that e1 AEðF1 Þ: We may assume that yx1 AEðF1 Þ: Let f2 ¼ xy1 : Applying (a)(ii) inductively to G2 we obtain a 2-factor F2 in G2 such that e2 ; f2 AEðF2 Þ: Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; eg is the required disconnected 2-factor of G: In order to verify (a)(ii) and (a)(iii), we label an edge of G1 as f1 : If f1 AEðH1 Þ we put f1 ¼ f : Otherwise f ¼ xi yi for some iAf1; 2; 3g: Relabelling if necessary, we may assume that f ¼ x1 y1 ; and put f1 ¼ x1 y: (a)(ii) Applying (a)(ii) inductively to G1 we obtain a 2-factor F1 in G1 such that e1 ; f1 AEðF1 Þ: We now construct F2 and F as in (a)(i). (a)(iii) Applying (a)(iii) inductively to G1 we obtain F1 in G1 such that dF1 ðuÞ ¼ dF1 ðyÞ ¼ 4; dF1 ðzÞ ¼ 2 for all zau; y; and e1 ; f1 are both contained in a common circuit C of F1 : We may assume that yx1 AEðCÞ: Let f2 ¼ xy1 : We now proceed as in (a)(i), but applying (a)(iii) inductively to G2 rather than (a)(ii), and putting F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; x2 y2 ; x3 y3 ; eg:
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Case 2: uAX -V ðH2 Þ and vAY -V ðH1 Þ: Let G1 be the multigraph obtained by adding an edge e1 with endvertices v; y to H1 ; and G2 be the multigraph obtained by adding an edge e2 with endvertices x; u to H2 : Since uAX we have uayi for 1pip3: Thus f AEðH2 Þ and we put f2 ¼ f : (a)(i) Applying (b)(ii) inductively to G2 we obtain a factor F2 in G2 such that dF2 ðxÞ ¼ 4; dF2 ðzÞ ¼ 2 for all zax; and e2 ; f2 are both contained in a common circuit C of F2 : We may assume that xy1 AEðCÞ: Let f1 ¼ yx1 : Applying (b)(i) inductively to G1 (with the roles of X and Y reversed) we obtain a factor F1 in G1 such that dF1 ðyÞ ¼ 4; dF1 ðzÞ ¼ 2 for all zay; and e1 ; f1 are both contained in a common circuit of F1 : Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; x2 y2 ; x3 y3 ; eg is the required disconnected 2-factor of G: (a)(ii) The disconnected 2-factor F constructed in (a)(i) contains e; f : (a)(iii) Applying (b)(i) inductively to G2 we obtain a factor F2 in G2 such that dF2 ðuÞ ¼ 4; dF2 ðzÞ ¼ 2 for all zau; and e2 ; f2 are both contained in a common circuit C of F2 : We may assume that xy1 AEðCÞ: Let f1 ¼ yx1 : Applying (b)(ii) inductively to G1 we obtain a factor F1 in G1 such that dF1 ðvÞ ¼ 4; dF1 ðzÞ ¼ 2 for all zav; and e1 ; f1 are both contained in a common circuit of F1 : Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; eg is the required factor of G: Case 3: uAX -V ðH1 Þ and vAX -V ðH2 Þ: Let G1 be the multigraph obtained by adding an edge e1 with endvertices u; y to H1 ; and G2 be the multigraph obtained by adding an edge e2 with endvertices x; v to H2 : In order to verify (b), we label an edge of G1 as f1 : If f1 AEðH1 Þ we put f1 ¼ f : Otherwise f ¼ xi yi for some iAf1; 2; 3g: Relabelling if necessary, we may assume that f ¼ x1 y1 ; and put f1 ¼ x1 y: (b)(i) Applying (a)(iii) inductively to G1 we obtain a factor F1 in G1 such that dF1 ðuÞ ¼ dF1 ðyÞ ¼ 4; dF1 ðzÞ ¼ 2 for all zau; y; and e1 ; f1 are both contained in a common circuit C of F1 : We may assume that yx1 AEðCÞ: Let f2 ¼ xy1 : Applying (b)(i) inductively to G2 we obtain a factor F2 in G2 such that dF2 ðxÞ ¼ 4; dF2 ðzÞ ¼ 2 for all zax; and e2 ; f2 are both contained in a common circuit of F2 : Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; x2 y2 ; x3 y3 ; eg is the required factor of G: (b)(ii) Applying (a)(ii) inductively to G1 we obtain a 2-factor F1 in G1 such that e1 ; f1 AEðF1 Þ: (Note that e1 ; f1 are necessarily contained in a common circuit of F1 since F1 is a 2-factor and e1 ; f1 are both incident to u:) We may assume that yx1 AEðF1 Þ: Let f2 ¼ xy1 : Applying (b)(ii) inductively to G2 we obtain a factor F2 in G2 such that dF2 ðvÞ ¼ 4; dF2 ðzÞ ¼ 2 for all zav; and e2 ; f2 are both contained in a common circuit of F2 : Then F ¼ ðF1 yÞ,ðF2 xÞ,fx1 y1 ; eg is the required factor of G: Case 4: uAX -V ðH2 Þ and vAX -V ðH1 Þ: Let G1 be the multigraph obtained by adding an edge e1 with endvertices v; y to H1 ; and G2 be the multigraph obtained by adding an edge e2 with endvertices x; u to H2 : Since uAX we have uayi for 1pip3: Thus f AEðH2 Þ and we put f2 ¼ f : (b)(i) We proceed in a similar fashion to Case 3 applying (b)(i) to G2 ; then (a)(ii) to G1 :
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(b)(ii) We proceed in a similar fashion to Case 3 applying (b)(ii) to G2 ; then (a)(iii) to G1 : & We may now prove our main result. Theorem 3.5. Let G be a 2-factor hamiltonian k-regular bipartite graph. Then kp3: Proof. Suppose that the theorem is false and let G be a counterexample chosen such that k is as small as possible. Let ðX ; Y Þ be the bipartition of G; where jX j ¼ jY j ¼ n: Let F be a 1-factor of G and H ¼ G F : Then H is a 2-factor hamiltonian ðk 1Þregular bipartite graph. Hence, by the minimality of k; we must have k ¼ 4: In addition, since H is cubic, it follows from Lemma 3.3, that jV ðHÞj 2 ðmod 4Þ and H is 3-connected. Let F 0 be a 1-factor of H: Since F ,F 0 is a Hamilton circuit of G and jV ðHÞj 2 ðmod 4Þ; we have sgnðF 0 Þ ¼ sgnðF Þ: Thus all 1-factors of H have the same sign and so H is det-extremal. Using Theorem 2.2 we deduce that HAH: We now obtain the required contradiction by applying Lemma 3.4(a)(i) to H; choosing e to be any edge of F : &
4. Minimally 1-factorable graphs Let G be a k-regular bipartite graph. We say that G is minimally 1-factorable if every 1-factor of G is contained in a unique 1-factorization of G: This paper was inspired by results on minimally 1-factorable graphs obtained in [1–4]. It can be seen that if G is minimally 1-factorable then G is 2-factor hamiltonian, and that, if k ¼ 2; 3; then G is minimally 1-factorable if and only if G is 2-factor hamiltonian. Using this observation, we could have deduced Lemma 3.3 from [3, Corollaries 2(iii) and 4]. Theorem 3.5 extends the result of [1] that minimally 1factorable k-regular bipartite graphs exist only when kp3: It follows from [2] that a smallest counterexample to Conjecture 3.2 is cubic and cyclically 4-edge connected, and from [4] that it has girth at least six. Thus, to prove the conjecture, it would suffice to show that the Heawood graph is the only 2-factor hamiltonian cyclically 4edge connected cubic bipartite graph of girth at least six.
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