Des. Codes Cryptogr. (2012) 64:153–160DOI 10.1007/s10623-011-9522-0

Irreducible pseudo 2-factor isomorphic cubic bipartitegraphs

Marién Abreu · Domenico Labbate · John Sheehan

Received: 29 September 2010 / Revised: 9 May 2011 / Accepted: 10 May 2011 /Published online: 28 May 2011© Springer Science+Business Media, LLC 2011

Abstract A bipartite graph is pseudo 2-factor isomorphic if the number of circuits in each2-factor of the graph is always even or always odd. We proved (Abreu et al., J Comb TheoryB 98:432–442, 2008) that the only essentially 4-edge-connected pseudo 2-factor isomorphiccubic bipartite graph of girth 4 is K3,3, and conjectured (Abreu et al., 2008, Conjecture3.6) that the only essentially 4-edge-connected cubic bipartite graphs are K3,3, the Heawoodgraph and the Pappus graph. There exists a characterization of symmetric configurations n3

due to Martinetti (1886) in which all symmetric configurations n3 can be obtained from aninfinite set of so called irreducible configurations (Martinetti, Annali di Matematica Pura edApplicata II 15:1–26, 1888). The list of irreducible configurations has been completed byBoben (Discret Math 307:331–344, 2007) in terms of their irreducible Levi graphs. In thispaper we characterize irreducible pseudo 2-factor isomorphic cubic bipartite graphs prov-ing that the only pseudo 2-factor isomorphic irreducible Levi graphs are the Heawood andPappus graphs. Moreover, the obtained characterization allows us to partially prove the aboveConjecture.

The authors would like to dedicate this paper to the loving memory of Lucia Gionfriddo.

This is one of several papers published together in Designs, Codes and Cryptography on the special topic:“Geometry, Combinatorial Designs & Cryptology”.

M. Abreu (B)Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata,C. da Macchia Romana, 85100 Potenza, Italye-mail: marien.abreu@unibas.it

D. LabbateDipartimento di Matematica, Politecnico di Bari, 70125 Bari, Italye-mail: labbate@poliba.it

J. SheehanDepartment of Mathematical Sciences, King’s College, Old Aberdeen AB24 3UE,Scotland, UKe-mail: j.sheehan@maths.abdn.ac.uk

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154 M. Abreu et al.

Keywords 2-factors · Irreducible configurations · Levi graphs

Mathematics Subject Classification (2000) 05C38 · 05C75 · 05B30 · 05C70

1 Introduction

All graphs considered are finite and simple (without loops or multiple edges). A graph witha 2-factor is said to be 2-factor hamiltonian if all its 2-factors are Hamilton circuits, and,more generally, 2-factor isomorphic if all its 2-factors are isomorphic. Examples of suchgraphs are K4, K5, K3,3, the Heawood graph (which are all 2-factor hamiltonian) and thePetersen graph (which is 2-factor isomorphic). Several recent papers have addressed theproblem of characterizing families of graphs (particularly regular graphs) which have theseproperties. It is shown in [2,6] that k-regular 2-factor isomorphic bipartite graphs exist onlywhen k ∈ {2, 3} and an infinite family of 3-regular 2-factor hamiltonian bipartite graphs,based on K3,3 and the Heawood graph, is constructed in [6]. It is conjectured in [6] that every3-regular 2-factor hamiltonian bipartite graph belongs to this family. Faudree et al. in [5]determine the maximum number of edges in both 2-factor hamiltonian graphs and 2-factorhamiltonian bipartite graphs. In addition, Diwan [4] has shown that K4 is the only 3-regular2-factor hamiltonian planar graph.

Moreover, 2-factor isomorphic bipartite graphs are extended in [1] to the more generalfamily of pseudo 2-factor isomorphic graphs i.e. graphs G with the property that the parityof the number of circuits in a 2-factor is the same for all 2-factors of G. Example of thesegraphs are K3,3, the Heawood graph H0 and the Pappus graph P0. Finally, it is proved in [1]that pseudo 2-factor isomorphic 2k-regular graphs and k-regular digraphs do not exist fork ≥ 4.

An incidence structure is linear if two different points are incident with at most one line.A symmetric configuration nk (or nk configuration) is a linear incidence structure consistingof n points and n lines such that each point and line is incident with k lines and points, respec-tively. Let C be a symmetric configuration nk , its Levi graph G(C ) is a k-regular bipartitegraph whose vertex set consists of the points and the lines of C and there is an edge betweena point and a line in the graph if and only if they are incident in C . We will also refer to Levigraphs of configurations as their incidence graphs.

Let G be a graph and E1 be an edge-cut of G. We say that E1 is a non-trivial edge-cut if allcomponents of G−E1 have at least two vertices. The graph G is essentially 4-edge-connectedif G is 3-edge-connected and has no non-trivial 3-edge-cuts.

Conjecture 1 [1] Let G be an essentially 4-edge-connected pseudo 2-factor isomorphic cubicbipartite graph. Then G ∈ {K3,3, H0, P0}.Theorem 1 [1] Let G be an essentially 4-edge-connected pseudo 2-factor isomorphic cubicbipartite graph. Suppose that G contains a 4-circuit. Then G = K3,3.

It follows from Theorem 1 that an essentially 4-edge-connected pseudo 2-factor isomor-phic cubic bipartite graph of girth greater than or equal to 6 is the Levi graph of a symmetricconfiguration n3. In 1886 V. Martinetti [9] characterized symmetric configurations n3, show-ing that they can be obtained from an infinite set of so called irreducible configurations, ofwhich he gave a list. Recently, Boben proved that Martinetti’s list of irreducible configurationswas incomplete and completed it [3]. Boben’s list of irreducible configurations was obtainedcharacterizing their Levi graphs, which he called irreducible Levi graphs (cf. Sect. 2).

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Irreducible pseudo 2-factor isomorphic cubic bipartite graphs 155

In this paper, we characterize irreducible pseudo 2-factor isomorphic cubic bipartite graphsproving that the Heawood and the Pappus graphs are the only irreducible Levi graphs whichare pseudo 2-factor isomorphic cubic bipartite. Moreover, the characterization obtainedallows us to partially prove Conjecture 1, i.e. in the case of irreducible pseudo 2-factorisomorphic cubic bipartite graphs.

2 Symmetric configurations n3

In 1886, Martinetti [9] provided a construction for a symmetric configuration n3 from asymmetric configuration (n − 1)3, say C . Suppose that in C there exist two parallel (non-intersecting) lines l1 = {α, α1, α2} and r1 = {β, β1, β2} such that the points α and β arenot on a common line. Then a symmetric configuration n3 is obtained from C by deletingthe lines l1, r1, adding a point μ and adding the lines h1 = {μ, α1, α2}, h2 = {μ, β1, β2}and h3 = {μ, α, β}. Not all symmetric configurations n3 can be obtained using this methodon some symmetric configuration (n − 1)3. The configurations that cannot be obtained inthis way are called irreducible configurations, while the others are reducible configurations.However, if all irreducible symmetric configurations n3 are known, then all symmetric con-figurations n3 can be constructed recursively with Martinetti’s method. The complete list ofirreducible configurations is given in [3, Theorem 8].

Theorem 2 The connected irreducible n3 configurations are:

1 cyclic configurations with base line {0, 1, 3};

2 the configurations with incidence graphs T1(n), T2(n), T3(n), n ≥ 1, respectively, eachof them giving precisely one (10n)3 configuration;

3 the Pappus configuration.

The Levi graphs of irreducible configurations, which are called irreducible Levi graphs.Such graphs turned out to be either the Pappus graph, or belong to one of four infinite familiesD(n), T1(n), T2(n), T3(n), n ≥ 1, which we now proceed to describe.

The D(n) family: Let C(n), n ≥ 1, be the graph on 6n vertices, consisting of n segments(6-circuits labeled as in Fig. 1), linked by the edges v1

i−1u1i , v

2i−1u4

i , and u3i−1u2

i , for i ≥ 2.Let the graph D(n), n ≥ 1 be defined as follows:For n ≡ 0 (mod 3) let D(n) be the graph C(m), m = n/3, with the edges u1

1v1m, u4

1v2m ,

and u21u3

m added (cf. Fig. 2).For n ≡ 1 (mod 3) let D(n) be the graph C(m), m = (n−1)/3, with two vertices w1

m, w2m

and the edges u11w

1m, u2

1v2m, u4

1w2m, w1

mw2m, w1

mu3m, w2

mv1m added.

Fig. 1 Label for the 6-circuits ofthe definition of C(n)

u1i

u2i

u3i

u4i

v1i

v2i

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156 M. Abreu et al.

Fig. 2 The graph D(9)

u11 v1

1 u12 v1

2 u13 v1

3

u21 u2

2 u23

u31 u3

2 u33

u41 v2

1 u42 v2

2 u43 v2

3

Fig. 3 Segment graph GTu1

i

w1i x1

i y1i z1

iv1

i

t1i t2

i

u2i

w2i

x2i

y2i z2

i v2i

u3i w3

i x3i y3

i z3i

v3i

For n ≡ 2 (mod 3) let D(n) be the graph C(m), m = (n − 2)/3, with four ver-tices w1

m, w2m, w3

m, w4m and the edges v1

mw1m, v2

mw4m, u3

mw2m, u1

1w4m, u2

1w1m, u4

1w3m, w1

mw2m,

w2mw3

m, w3mw4

m added.A cyclic configuration has Zn = {0, 1, . . . , n − 1} as set of points and B ={{0, b, c},

{1, b + 1, c + 1}, . . . , {n − 1, b + n − 1, c + n − 1}} as set of lines, where the operations aremodulo n, and the base line is {0, b, c} for b, c ∈ Zn .

Note that the graphs D(n) are the Levi graphs of the cyclic n3 configurations with baseline {0, 1, 3}. In particular, for n = 7 the cyclic 73 configuration is the Fano plane and D(7)

is the Heawood graph H0.The T1(n), T2(n) and T3(n) families: Let T (n), n ≥ 1, be the graph on 20n vertices con-

sisting of n segments GT shown in Fig. 3, linked by the edges v1i−1u1

i , v2i−1u2

i , v3i−1u3

i , fori ≥ 2.

Let T1(n) be the graph obtained from T (n) by adding the edges u11v

1n, u2

1v2n, u3

1v3n . Let

T2(n) be the graph obtained from T (n) by adding the edges u31v

1n, u2

1v2n, u1

1v3n . Let T3(n) be

the graph obtained from T (n) by adding the edges u11v

3n, u2

1v1n, u3

1v2n . In [3], Boben proved

that for each fixed value of n, no two of the graphs T1(n), T2(n), T3(n) are isomorphic.Note that T1(1) is the Levi graph of Desargues’ configuration, and T2(1), T3(1) correspond

to the Levi graphs of the configurations 103 F and 103G respectively according to Kantor’snotation for the ten 103 configurations described in [8].

The Pappus graph: Recall that the Levi graph of the Pappus 93 configuration is the follow-ing pseudo 2-factor isomorphic but not 2-factor isomorphic cubic bipartite graph [1], calledthe Pappus graph P0.

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Irreducible pseudo 2-factor isomorphic cubic bipartite graphs 157

v1 v2v3

v4

v5

v6

v7

v8

v9v10v11

v12

v13

v14

v15

v16

v17

v18

3 2-Factors of irreducible Levi graphs

Let G be a graph and u, v be two vertices in G. Then a (u, v)-path is a path from u to v.Given two disjoint paths P = u1, . . . , un and Q = un+1, . . . , un+m (except, possibly, foru1 = un+m), the path P Q = u1, . . . , un+m is the concatenation of P and Q together withthe edge unun+1. Similarly, for a vertex v ∈ (G − P) ∪ {u1} the path Pv is composed byP, v and the edge unv. If u1 = un+m or u1 = v we write (P Q) and (Pv) respectively, toemphasize that P Q and Pv are circuits.

Theorem 3 The Heawood and the Pappus graphs are the only irreducible Levi graphs whichare pseudo 2-factor isomorphic.

Proof It is straightforward to show that the Heawood graph H0 is 2- factor hamiltonian andhence pseudo 2-factor isomorphic (cf. [6]). We have already proved in [1, Proposition 3.3]that the Pappus graph is pseudo 2-factor isomorphic. We need to prove that all other irreduc-ible Levi graphs are not pseudo 2-factor isomorphic and we will do so by finding in each ofthem a 2-factor with an odd number of circuits and another 2-factor with an even number ofcircuits.

The following paths will be used for constructing 2-factors in D(n), for n ≥ 8.

L1i = u1

i u2i u3

i u4i v

2i v1

i L2i = u4

i u3i u2

i u1i v

1i v2

iM1 = u4

1v21u4

2u32u2

3u33u4

3v22v1

2u13v

13v2

3 M2 = u4mv2

mv1mu1

mu2mu3

mw1mw2

mu41

Ni = u2i u1

i v1i−1v

2i−1u4

i u3i Nm = w2

mw1mv1

mv2mw4

mw3mu4

1v21u4

2u32

C1 = (u11u2

1u31u2

2u12v

11u1

1) C2 = (u11u2

1u31u4

1w2mw1

mu11)

C3 = (v11v2

1u42v

22v1

2u13v

13v2

3u43u3

3u23u3

2u22u1

2v11)

Hamiltonian 2-factors in D(n) are

⎧⎨

⎩

(L11L1

2 · · · L1mu1

1) n ≡ 0 mod 3(L1

1L12 · · · L1

mw2mw1

mu11) n ≡ 1 mod 3

(L11L1

2 · · · L1mw1

mw2mw3

mw4mu1

1) n ≡ 2 mod 3Disconnected 2-factors with exactly two circuits in D(n) are

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

C1 ∪ (M1u41) n = 9

for n ≡ 0 mod 3C1 ∪ (M1L2

4 · · · L2mu4

1) n = 3m, m ≥ 4

C2 ∪ C3 n = 10for n ≡ 1 mod 3C1 ∪ (M1 M2) n = 13

C1 ∪ (M1L24 · · · L2

m−1 M2) n = 3m + 1, m ≥ 5

C1 ∪ (Nmw2m) n = 8

for n ≡ 2 mod 3C1 ∪ (N3 · · · Nmu2

3) n = 3m + 2, m ≥ 3

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158 M. Abreu et al.

Now we shall find such pairs of 2-factors for the graphs T1(n), T2(n) and T3(n), n ≥ 1. Tothis purpose we consider the following six paths in the segment graph GT from Fig. 3.

P1i = u2

i , w1i , u1

i , w2i , u3

i , w3i , x3

i , y3i , z3

i , v3i , t2

i , v1i , z1

i , y2i , x2

i , t1i , x1

i , y1i , z2

i , v2i

P2i = u2

i , w1i , u1

i , w2i , u3

i , w3i , x3

i , y3i , z2

i , v2i

(P3i ) = (v1

i , t2i , v3

i , z3i , y2

i , x2i , t1

i , x1i , y1

i , z1i , v

1i )

Q1i = u3

i , w3i , u2

i , w1i , u1

i , w2i , x2

i , y2i , z3

i , v3i , t2

i , v1i , z1

i , y1i , x1

i , t1i , x3

i , y3i , z2

i , v2i

Q2i = u3

i , w2i , u1

i , w1i , u2

i , w3i , x3

i , y3i , z2

i , v2i

P1i P2i and P3i

Q1i Q2i and P3i

The paths P1i and Q1

i are hamiltonian (u2i , v

2i ) and (u3

i , v2i )-paths, respectively. The paths

P2i and Q2

i are (u2i , v

2i ), and (u3

i , v2i )-paths on 10 vertices, respectively. Finally, (P3

i ) is a

10-circuit in GT − P2i = GT − Q2

i .In T1(n) and T2(n) the hamiltonian 2-factor F1(n) = (P1

1 P12 · · · P1

n u21) and the discon-

nected 2-factor F2(n) = (P21 P1

2 · · · P1n u2

1) ∪ (P31 ), which consists of exactly two circuits,

show that these graphs are not pseudo 2-factor isomorphic.Similarly, in T3(n) the hamiltonian 2-factor F ′

1(1) = (Q11 P1

2 · · · P1n u3

1) and the discon-nected 2-factor F ′

2(1) = (Q21 P1

2 · · · P1n u3

1) ∪ (P31 ), which consists of exactly two circuits,

show that this graph is not pseudo 2-factor isomorphic.

Note that Theorem 3 proves Conjecture 1 in the case of irreducible pseudo 2-factor iso-morphic cubic bipartite graphs. In the next section we show that Conjecture 1 cannot beproved directly from Theorem 3 by extending it to reducible Levi graphs.

4 2-Factors in extensions and reductions of Levi graphs of n3 configurations

Recall that a Martinetti extension can be described in terms of graphs as follows:Let G1 be the Levi graph of a symmetric configuration n3 and suppose that in G1 there

are two edges e1 = x1 y1 and e2 = x2 y2 such that neither x1, x2 nor y1, y2 have a commonneighbour, then the graph G := G1 −{e1, e2}+{u, v}+{ux1, ux2, vy1, vy2, uv}, where u, v

are new vertices, is the Levi graph of an (n + 1)3 configuration (Fig. 4).

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Irreducible pseudo 2-factor isomorphic cubic bipartite graphs 159

Fig. 4 Martinetti extension

y1 y2

x1 x2

y1 y2

v

u

x1 x2

Fig. 5 Martinetti reduction

y1 y2

v

u

x1 x2

e

y1 y2

x1 x2

or

y1 y2

x1 x2

Similarly the Levi graph G of a symmetric configuration (n + 1)3 is Martinetti reduc-ible if there is an edge e = uv in G such that either G := G1 − {u, v} + x1 y1 + x2 y2 orG := G1 − {u, v} + x1 y2 + x2 y1 is again the Levi graph of a symmetric configuration n3,where x1, x2, y1, y2 are the neighbours of u and v as in Fig. 5.

It is well known that the 73 configuration, whose Levi graph is the Heawood graph, is notMartinetti extendible and that the Pappus configuration is Martinetti extendible in a uniqueway; it is easy to show that this extension is not pseudo 2-factor isomorphic.

Let C be a symmetric configuration n3 and C ′ be a symmetric configuration (n + 1)3

obtained from C through a Martinetti extension. It can be easily checked that there are 2-factors in C ′ that cannot be reduced to a 2-factor in C . For example if C corresponds to thefirst option in Fig. 5, a 2-factor of C ′ containing the path x1uvy2 will not reduce to a 2-factorin C . Conversely, there might be 2-factors of C for which the parity of the number of circuitsis not preserved when extended to a 2-factor in C ′. For example, consider the graph H0 ∗ H0,the star product of the Heawood graph with itself (see [7, p. 90]). This graph is 2-factorhamiltonian and Martinetti reducible (only through the edges of the non-trivial 3-edge-cut).On the other hand, all of its Martinetti reductions are no longer pseudo 2-factor isomorphic.One could have hoped for a proof of Conjecture 1 based on the study of the 2-factors ofreducible configurations, once the set of 2-factors of the underlying irreducible configura-tions is determined to some extent. The previous discussion shows that this approach is notfeasible.

References

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2. Aldred R., Funk M., Jackson B., Labbate D., Sheehan J.: Regular bipartite graphs with all 2-factorsisomorphic. J. Comb. Theory B 92(1), 151–161 (2004).

3. Boben M.: Irreducible (v3) configurations and graphs. Discret. Math. 307, 331–344 (2007).4. Diwan A.A.: Disconnected 2-factors in planar cubic bridgeless graphs. J. Comb. Theory B 84, 249–259

(2002).5. Faudree R.J., Gould R.J., Jacobson M.S.: On the extremal number of edges in 2-factor hamiltonian graphs.

In: Graph Theory—Trends in Mathematics, pp. 139–148, Birkhäuser, Basel (2006).6. Funk M., Jackson B., Labbate D., Sheehan J.: 2-Factor hamiltonian graphs. J. Comb. Theory B 87(1),

138–144 (2003).

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7. Holton D.A., Sheehan J.: The Petersen graph. In: Australian Mathematical Society Lecture Series, vol.7. Cambridge University Press, Cambridge (1993).

8. Kantor S.: Die configurationen (3, 3)10, Sitzungsber. Wiener Akad. 84, 1291–1314 (1881).9. Martinetti V.: Sulle configurazioni piane μ3. Annali di Matematica Pura ed Applicata II 15, 1–26

(dall’aprile 1867 al gennaio 1888).

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