On the perfect matching index of bridgeless cubic graphs

ON THE PERFECT MA TCHING INDEX OF BRIDGELESS CUBIC GRAPHS J.L. FOUQUET AND J.M. V ANHERPE Abstract. If G is a bridgeless cubic graph, F ulkerson conjec tured that we can find 6 p erfect matchings M 1 , . . . , M 6 of G with the property that ev ery edge of G is con tained in exactly tw o of them and Berge conjectured that its edge set can be co v ered by 5 p erfect matchings. W e define τ ( G ) as the least n umber of p erfect matchings allowing to cov e r the edge set of a bridgeless cubic graph and we study this parameter. The set of graphs wi th p erfect matc h ing index 4 seems inte resting and we give some informations on this class. 1. Introduction The following conjecture is due to F ulkerson, and app ears fir st in [6]. Conjecture 1.1. If G is a bridgel ess cubic gr ap h, then ther e exist 6 p erfe ct match- ings M 1 , . . . , M 6 of G with the pr op ert y that every e dge of G is c ontaine d in exactly two of M 1 , . . . , M 6 . If G is 3- edge-colour able, then we ma y choose thre e p er fect matc hings M 1 , M 2 , M 3 so that every edge is in exa ctly one. T aking e a ch of thes e twice gives us 6 p erfect matchings with the prop er ties describ ed ab ove. Th us, the ab ov e conjecture holds trivially for 3- edge-color able graphs. There do exis t bridgeless cubic g r aphs which are not 3 − edge-co lourable (for instance the Petersen gra ph), but the a bove conjec- ture asserts that every such graph is close to b eing 3 -edge-colo urable. If F ulkerson’s conjecture were true, then deleting one of the p erfect matchings from the double cover would result in a co vering of the g raph by 5 p erfect matc hings. This weak er conjecture was prop osed by Berge (see Seymour [12]). Conjecture 1.2. If G is a bridgeless cubic gr aph, then ther e exists a c overing of its e dges by 5 p erfe ct matchings. Since the Petersen gra ph do es not admit a covering by les s that 5 p erfect match- ings (see section 3), 5 in the ab ove co njecture ca n not b e changed into 4 and the following weakening of conjecture 1.2 (sugg ested b y Berge) is still op en. Conjecture 1.3. Ther e exists a fi xe d inte ger k su ch that the e dge set of every bridgeless cubic gr aph c an b e written as a u n ion of k p erfe ct matchings. Another c o nsequence of the F ulkerson conjecture w ould b e that every bridg eless cubic gra ph has 3 pe r fect matchings with empt y intersection (take any 3 o f the 6 per fect matchings given by the conjecture). The following weak e ning of this (also suggested by Be rge) is still op en. 1991 Mathematics Subje ct Classific ati on. 035 C. Key wor ds and phr ases. Cubic graph; Perfect M atc hings; 1 2 J.L. FOUQUET AND J.M. V ANHERPE Conjecture 1.4. Ther e ex ists a fixe d inte ger k such that every bridgeless cubic gr aph has a list of k p erfe ct matchings with empty interse ction. F o r k = 3 this conjecture is known as the F an Ras pa ud Conjecture. Conjecture 1.5. [3] Every bridgeless cubic gr aph c ontains p erfe ct matching M 1 , M 2 , M 3 such that M 1 ∩ M 2 ∩ M 3 = ∅ While some par tial res ults e x ist c o ncerning conjecture 1.5 (se e [1 7]), we hav e noticed no r esult in the liter ature conc e rning the v alidity of C o njecture 1.1 or Conjecture 1 .4 for te usual classes o f graphs which a r e examined when dealing with the 5 − flow co njectur e of T utte [15] or the cycle do uble conjectur e of Seymo ur [11] and Szekeres [13]. Hence for bridgeless cubic gra phs with o ddness 2 (a 2 − factor contains exactly tow o dd cycles) it is kno wn that the 5 − flow conjecture holds true as w ell as the c ycle do uble conjecture (see Zhang [18] for a comprehensive s tudy of this sub ject). Let G b e a br idg eless cubic gr aph, we shall say that the set M = { M 1 , . . . M k } ( k ≥ 3) of p erfect matchings is a k − c overing when ea ch edge is contained in a t least one of theses pe rfect ma tc hings. A F ulkerson c overing is a 6 − cov er ing where each edge app ea r s exa ctly twice. Since ev er y edge of a bridgeless cubic g raph is contained in a p erfect matching (see [1 0]) the minimum num b er τ ( G ) o f p erfect matchings cov er ing its edge set is well defined. W e shall sa y that τ ( G ) is the p erfe ct matching index of G . W e o b viously have that τ ( G ) = 3 if a nd only if G is 3 − edge- colourable. 2. Preliminaries resul ts Prop ositio n 2 .1. let G b e a cubic gr aph with a k − c overing M = { M 1 , . . . , M k } ( k ≥ 3 ) then G is bridgeless. Pro of Assume that e ∈ E ( G ) is an isthmus, then the edges incident to e a re not cov er e d b y an y p erfect matching of G a nd M is no t a K − cov ering, a contradiction.  2.1. 2 − cut connection. Let G 1 , G 2 be t wo bridgeless cubic g raph and e 1 = u 1 v 1 ∈ E ( G 1 ), e 2 = u 2 v 2 ∈ E ( G 1 ) b e tw o edges. Construct a new graph G = G 1 J G 2 G = [ G 1 \ { e 1 } ] ∪ [ G 2 \ { e 2 } ] ∪ { u 1 u 2 , v 1 v 2 } Prop ositio n 2.2. L et G 1 b e a cu bic gr aph such that τ ( G 1 ) = k ≥ 3 and let G 2 b e any cubic bridgeless gr aph, t hen τ ( G 1 J G 2 ) ≥ k Pro of Let G = G 1 J G 2 . Assume that k ′ = τ ( G ) < k and let M = { M 1 , . . . , M k ′ } be a k ′ − cov e r ing of G Any p erfect ma tc hing o f G must intersect the 2 − edge cut { u 1 v 1 , u 2 v 2 } in tw o edge s or ha s no e dge in common with that set. T hus any p er- fect ma tc hing in M leads to a p erfect matching of G 1 . Hence we s ho uld hav e a k ′ − cov e r ing o f the edge set o f G 1 , a contradiction.  3 2.2. 3 − cut connection. Let G 1 , G 2 be tw o bridgele ss cubic gra ph a nd u ∈ V ( G 1 ), v ∈ V ( G 2 ) b e tw o vertices with N ( u ) = { u 1 , u 2 , u 3 } and N ( v ) = { v 1 , v 2 , v 3 } . Construct a new g raph G = G 1 ⊗ G 2 G = [ G 1 \ { u } ] ∪ [ G 2 \ { v } ] ∪ { u 1 v 1 , u 2 v 2 , u 3 v 3 } It is well known that the resulting gr aph G 1 ⊗ G 2 is bridgeless . The 3 − edge cut { u 1 v 1 , u 2 v 2 , u 3 v 3 } will b e ca lled the princip al 3 − e dge cut . Prop ositio n 2.3. L et G 1 b e a cu bic gr aph such that τ ( G 1 ) = k ≥ 3 and let G 2 b e any cubic bridgeless gr aph. L et k ′ = τ ( G 1 ⊗ G 2 ) and let M = { M 1 , . . . , M k ′ } b e a k ′ − c overing of G 1 ⊗ G 2 . Then one of the fol lowings is true (1) k ′ ≥ k (2) Ther e is a p erfe ct matching M i ∈ M ( 1 ≤ i ≤ k ) c ontaining the princip al 3 − e dge cut Pro of Assume that k ′ < k . An y p erfect matching o f G 1 ⊗ G 2 m ust intersect the principal 3 − edge cut in one or three edg es. If none of the p er fect matchings in M contains the principal 3 − edge cut, then a n y perfect matching in M leads to a p erfect matching of G 1 and any edge of G 1 is covered by one o f these pe r fect matchings. Hence we should ha ve a k ′ − cov e r ing of the edge set of G 1 , a contradic- tion.  3. On graphs with perfect ma tching index 4 A na tur al question is to inv estigate the cla ss o f graphs for which the p erfect matching index is 4. Prop ositio n 3.1. L et G b e a cubic gr aph with a 4 − c overing M = { M 1 , M 2 , M 3 , M 4 } then (1) Every e dge is c ontaine d in exactly one or two p erfe ct matchings of M . (2) The set M of e dges c ont aine d in exactly two p erfe ct matchings of M is a p erfe ct matching. (3) If τ ( G ) = 4 then ∀ i 6 = j ∈ { 1 , 2 , 3 , 4 } M i ∩ M j 6 = ∅ . Pro of Let v b e any vertex of G , each edg e incident with v must be c ontained in so me p erfect ma tc hing of M and e a ch p erfect matching must b e inciden t with v . W e have thus exa ctly one edge incident with v which is cov e r ed by exactly tw o per fect ma tc hings of M while the t wo o ther edges a re covered b y exactly o ne p er fect matching. W e get thus immediately Items 1 and 2. When τ ( G ) = 4, G is not a 3 − edge c o lourable graph. Ass ume that we hav e tw o per fect matchings with a n empty intersection. These tw o p erfect matc hing s lea d to an even 2 − factor and hence a a 3 − edge colour ing of G , a contradiction.  In the fo llowing the edges of the matching M described in item 2 of Prop os itio n 3.1 will b e said to b e c over e d t wic e . Prop ositio n 3.2. L et G b e a cubic gr aph such t hat τ ( G ) = 4 then G has at le ast 12 vertic es Pro of Let M = { M 1 , M 2 , M 3 , M 4 } be a cov er ing o f the edg e set of G into 4 p er - fect matchings. F ro m Pro p os ition 3.1 w e must hav e at lea st 6 edges in the p erfect matching formed with the edges covered twice in M . Hence, G must hav e at least 4 J.L. FOUQUET AND J.M. V ANHERPE 12 vertices as claimed.  F r om Prop ositio n 3.2, w e o bviously hav e that the Petersen graph has a p erfect matching index e qual to 5. Prop ositio n 3.3. L et G b e a cubic gra ph such tha t τ ( G ) = 4 and let M = { M 1 , M 2 , M 3 , M 4 } b e a c overing of its e dge set int o 4 p erfe ct matchings then for e ach j ( j = 1 . . . 4 ) M − M j is a set of 3 p erfe ct matchings satisfying the F an R asp aud c onje ctu r e. Pro of Obvious since, by Item 1 of Prop osition 3.1 any edge is contained in exactly one or tw o p erfect matchings of M .  Let G b e a cubic g raph with 3 p erfect matchings M 1 , M 2 and M 3 having an empt y intersection. Since such a gr aph satisfy the F an Ras paud c o njecture, when considering these thr ee p erfect ma tch ings, we s hall say that ( M 1 , M 2 , M 3 ) is a n FR-triple . When a cubic graph has a FR-triple we define T i ( i = 0 , 1 , 2) as the set of edges that b elong to prec isely i matchings of the FR-triple . Thus ( T 0 , T 1 , T 2 ) is a partition of the edg e set. Prop ositio n 3.4. L et G b e a cubic gr aph with 3 p erfe ct matchings M 1 , M 2 and M 3 having an empty interse ction. Then the set T 0 ∪ T 2 is a set of disjoint even cycles. Mor e over, the e dges of T 0 and T 2 alternate along these cycles. Pro of Let v be a vertex incident to a edge of T 0 . Since v must b e incident to each per fect matching and since the three p erfect matchings hav e a n empty in tersection, one of the r emaining edg e s incident to v m us t b e contained into 2 p erfect matchings while the other is contained in exactly o ne p erfect matc hing. The result follows.  Let G be a bridgeless cubic gra ph and let C and C ′ be distinct o dd cycles of G . Assume that there a re three distinct edges na mely xx ′ , y y ′ and z z ′ such that x , y and z a re vertices of C while x ′ , y ′ , z ′ are v er tices of C ′ which determine on C and on C ′ edge-disjoint paths o f o dd leng th then we shall say that ( xx ′ , y y ′ , z z ′ ) is a go o d triple and that the pair of cycles { C , C ′ } is a go o d p air . Theorem 3.5. L et G b e a cubic gr aph which has a 2 − factor F whose o dd cycles c an b e arr ange d into go o d p airs { C 1 , D 1 } , { C 2 , D 2 } , ..., { C k , D k } . Then τ ( G ) ≤ 4 . Pro of F or each go o d pair { C i , D i } let ( c 1 i d 1 i , c 2 i d 2 i , c 3 i d 3 i ) b e a go o d tr iple o f C i and D i , c 1 i , c 2 i , c 3 i being vertices of C i while d 1 i , d 2 i and 3 i are on D i . In o rder to construct a set M = { M 1 , M 2 , M 3 , M 4 } of 4 p erfect matchings covering the edg e set of G we let M 1 as the per fect matching of G obtained by deleting the edge s of the 2 − facto r. Let A j be the set of edges { c j i d j i | i = 1 . . . k } . W e construct a per fect matc hing M j ( j = 2 , 3 , 4) of G s uch that M 1 ∩ M j = A j . F o r each g o o d pair { C i , D i } ( i = 1 . . . k ), we a dd to A j the unique p erfect matching co ntained in E ( C i ) ∪ E ( D i ) when the t wo vertices c j i and d j i are deleted. W e g et hence 3 ma tc hings B j ( j = 2 , 3 , 4) where each vertex co nt ained in a go o d pair is sa turated. If the 2 − factor contains s ome even c y cles, we add first a p erfect matching contained in the edge set of these even cycles to B 2 . W e obtain th us a p erfect matching M 2 whose in terse c tion with M 1 is reduced to A 2 . The re ma ining edges of these even cycles are a dded to B 3 and to 5 B 4 , leading to the perfect matchings M 3 and M 4 . Let us remar k that each edge of these even cy cles are contained in M 2 ∪ M 3 . W e claim that each edge of G is contained in at least one of M = { M 1 , M 2 , M 3 , M 4 } . Since M 1 is the per fect matc hing which complements in G the 2 − factor F , the above remark s ays that we hav e just to prove that ea ch edg e of each go o d pair is cov- ered by s ome p erfect matching of M . By construction, no edge is c o nt ained in M 1 ∪ M 2 ∪ M 3 which means that ( M 1 , M 2 , M 3 ) is an FR-triple. In the sa me way , ( M 1 , M 3 , M 4 ) and ( M 1 , M 2 , M 4 ) ar e FR-triples. The edges o f T 0 ∪ T 2 induced by the FR-triple ( M 1 , M 2 , M 3 ) on each go o d pair { C i , D i } is the even cycle Γ i using c 1 i d 1 i and c 2 i d 2 i , the o dd pa th of C i joining c 1 i to c 2 i and the o dd pa th of D i joining d 1 i to d 2 i . In the same way , e dges o f T 0 ∪ T 2 induced b y the FR-triple ( M 1 , M 3 , M 4 ) on each go o d pair { C i , D i } is the even cycle Λ i using c 2 i d 2 i and c 3 i d 3 i , the o dd path of C i joining c 2 i to c 3 i and the o dd path of D i joining d 2 i to d 3 i . It is an easy ta s k to see that thes e tw o cycles Γ i and Λ i hav e the only e dge c 2 i d 2 i in co mmon. Hence each edge of Γ i ∩ T 0 is co nt ained into M 4 while ea ch edge o f Λ i ∩ T 0 is contained int o M 2 . The result follows.  3.1. On balanced matc hi ngs. A s et A ⊆ E ( G ) is a b alanc e d matching when w e can find 2 p erfect matc hings M 1 and M 2 such that A = M 1 ∩ M 2 . L e t B ( G ) be the set o f balance d matchings o f G , we define b ( G ) as the minimu m size of a any set A ∈ B ( G ), we ha ve: Prop ositio n 3.6. L et G b e a cu bic gr aph such that τ ( G ) = 4 then b ( G ) ≤ n 12 . Pro of Let M = { M 1 , M 2 , M 3 , M 4 } b e a cov er ing of the edge set of G into 4 per fect matc hings and let M b e the perfect matching of edges co n tained in exactly t wo p erfect matchings of M (Iem 2 o f Pr o po sition ?? ). Since M i ∩ M j 6 = ∅ ∀ i 6 = j ∈ { 1 , 2 , 3 , 4 } b y P rop osition ?? , these 6 bala nced matc hings partition M . Hence, one of them must ha ve a t most | M | 6 = n 12 edges.  In [14] Ka iser, Kr´ al and Norine proved Theorem 3. 7. Any bri dgeless cubic gr aph c ontains 2 p erfe ct matchings whose union c over at le ast 9 n 10 e dges of G . F r om Theor em 3.7, we can find t wo p erfect match ings with an intersection having at most n 10 edges in any cubic br idgeless graph. It can b e proved (see [4] ) that for any cyc lic a lly 4-edge connected cubic g raph G , either b ( G ) ≤ n 14 or any p er fect matching c o nt ains an o dd cut of size 5. 3.2. On classical snarks. As usual a snark is a non 3 − edge colourable bridgeless cubic g raph. In Figure 1 is depicted one of the two the Blanu ˇ sa snark s o n 1 8 v er tices [1]. In b old we hav e drawn a 2 − factor (each cycle has length 9) and the dashed edges connect the tr iple ( x, y , z ) of o ne cyc le to the triple ( x ′ , y ′ , z ′ ) of the second cycle. It is a routine matter to chec k that ( xx ′ , y y ′ , z z ′ ) is a go o d triple and Theorem 3.5 allows us to say that this graph has p erfect matc hing index 4. In the same wa y the second Blanu ˇ sa snark on 18 vertices depicted in Figure 2 can b e cov er e d b y 4 per fect matchings by using Theor em 3.5. F o r an o dd k ≥ 3 the Flower Sna rk F k int o duced by Is aac (see [8]) is the c ubic graph on 4 k vertices x 0 , x 1 , . . . x k − 1 , y 0 , y 1 , . . . y k − 1 , z 0 , z 1 , . . . z k − 1 , t 0 , t 1 , . . . t k − 1 6 J.L. FOUQUET AND J.M. V ANHERPE x z’ y’ x’ z y Figure 1 . Blanu ˇ sa snar k #1 x z z’ y y’ x’ Figure 2 . Blanu ˇ sa snar k #2 such that x 0 x 1 . . . x k − 1 is an induced cycle of length k , y 0 y 1 . . . y k − 1 z 0 z 1 . . . z k − 1 is an induced cycle o f length 2 k and for i = 0 . . . k − 1 the vertex t i is adjacent to x i , y i and z i . The set { t i , x i , y i , z i } induces the claw C i . In Figure 3 we hav e a representation of F 5 , the half edges (to the left and to the righ t in the figure) with same lab els are identified. a z4 t4 y4 z3 t3 y3 z0 t0 y0 z1 t1 y1 z2 y2 t2 x2 x1 x0 b c a c b Figure 3 . J 5 Theorem 3.8. τ ( F k ) = 4 . Pro of Let k = 2 p + 1 ≥ 3 and let C = x 0 x 1 . . . x 2 p , D = y 0 t 0 z 0 z 1 t 1 y 1 . . . y 2 i t 2 i z 2 i z 2 i +1 t 2 i +1 y 2 i +1 . . . y 2 p t 2 p z 2 p (0 ≤ i ≤ ) b e the o dd c y cles of lengths 2 k + 1 and 3 × (2 k + 1) res pectively which partition F k (in b old in Figure 3 . It is a routine matter to chec k that the edges x 0 t 0 , x 1 t 1 and x 2 t 2 form a g o o d tr iple (dashed edges 7 in Fig ure 3). Hence ( C, D ) is a 2 − factor of G and it is a go o d pair . The res ult follows fro m Theo rem 3.5.  Let H b e the g raph depicted in Figure 4 a c h g f e d b Figure 4 . H Let G k ( k o dd) b e a cubic graph o btained from k copies of H ( H 0 . . . H k − 1 where the name o f vertices are indexed b y i ) in a dding edges a i a i +1 , c i c i +1 , e i e i +1 , f i f i +1 and h i h i +1 (subscripts are taken mo dulo k ). If k = 5, then G k is known as the Go ldber g snark. Accordingly , we re fer to all graphs G k as Goldb erg graphs. The graph G 5 is shown in Figure 5 . The half edges (to the left a nd to the rig ht in the figur e) with same lab els are iden tified. x y z y x z Figure 5 . Goldb erg sna rk G 5 Theorem 3.9. τ ( G k ) = 4 . Pro of Let k = 2 p + 1 ≥ 3 and let C = a 0 a 1 . . . a 2 p , D = e 0 d 0 b 0 g 0 f 0 e 1 d 1 b 1 g 1 f 1 . . . e i d i b i g i f i . . . e 2 p d 2 p b 2 p g 2 p f 2 p (0 ≤ i ≤ ) b e the o dd cycles o f lengths 2 k + 1 and 5 × (2 k + 1) resp ectively and E = c 0 h 0 c 1 h 1 . . . c i h i . . . c 2 p h 2 p the cycle of length 4 k of G k . This set o f 3 cycles is a 2 − fa ctor of G k (in b old in Figure 5). At last, a 0 b 0 , a 1 b 1 and a 2 b 2 are edges of G (dashed edges in Figure 5). Then ( a 0 b 0 , a 1 b 1 , a 2 b 2 ) is a go o d triple. Hence ( C, D , E ) is a 2 − factor of G where ( C, D ) is a go o d pair . The result follows from Theorem 3.5.  3.3. On p ermutation graphs. A cubic gra ph G is called a p ermutation gr aph if G has a 2 − factor F such that F is the union of tw o ch ordless cycles C and C ′ . Let M b e the per fect ma tc hing G − F . A subgraph homeomorphic to the Petersen graph with no edge o f M subdiv ide d is calle d a M − P 10 . Ellingham [2] show ed that a p ermutation graph without any M − P 10 is 3 − edge colour a ble. 8 J.L. FOUQUET AND J.M. V ANHERPE 1 c ′ b ′ d ′ a ′ c d b a odd even C ′ C 1 1 1 Figure 6. The cy c le on 8 vertices describ ed in Lemma 3 .11 In genera l, w e do not k now whether a p ermutation graph distinct fro m the Petersen gr aph is 3 − edge colourable o r not. It is an easy task to construct a cyclically 4 − edge connected pe r mut ation graph which is a sna rk (consider the t wo Blanusa sna rks on 18 vertices for exemple) and Zha ng [18] conjectured: Conjecture 3.10 . L et G b e a 3 − c onne cte d cyclic al ly 5 − e dge c onne cte d p ermut a- tion gr aph. If G is a sn ark, t hen G must b e the Petersen gr aph. Let us co nsider a p ermutation gr a ph G with a 2-factor F having tw o cy cles C and C ′ . Two distinct vertices of C say x a nd y determine on C tw o paths with x and y as end- p oints. In or der to be unambiguous when co nsidering tho se paths from their end-p oints w e give an orientation to C . Thus C ( x, y ) will denote in the following the pa th of C that starts with the vertex x and ends with the vertex x according to the or ient ation of C .The notatio n C ′ ( x ′ , y ′ ) is defined similarly when x ′ and y ′ are vertices of C ′ . In order to determine which pe r mut ation graphs have a perfect ma tc hing index less than 4 we state the following too l (see Figure 6) : Lemma 3. 11. L et G b e a p ermutation gr aph with a 2 -factor c ontaining pr e cisely two o dd cycles C and C ′ . Assume that χ ′ ( G ) = 4 and that ( C , C ′ ) is not a go o d p air. Le t ab b e an e dge of C such t hat the o dd p ath determine d on C ′ with the neighb ors of a and b , say a ′ and b ′ r esp e ctively, has minimum length. Ass u me that C and C ′ have an orientation such that C ( a, b ) is an e dge and C ′ ( a ′ , b ′ ) has o dd length. Then ther e must ex ist 4 additional vertic es c and d on C and their n eighb ors on C ′ , say c ′ and d ′ r esp e ctively, verifying : • the p aths C ′ ( a ′ , d ′ ) , C ′ ( b ′ , c ′ ) and C ( d, c ) ar e e dges. • the p ath C ( b, d ) is o dd and the p ath C ′ ( d ′ , b ′ ) is even. Pro of Observe first that a ′ and b ′ are not a djacent other wise the c y cle obtained with the paths C ( b , a ) and C ′ ( b ′ , a ′ ) together with the edges aa ′ and bb ′ would be hamiltonian, a contradiction since it is as s umed that χ ′ ( G ) = 4 . Since the path C ′ ( a ′ , b ′ ) is o dd ther e must b e a neig h bo r of b ′ on C ′ ( b ′ , a ′ ), say c ′ . Let c b e the neighbor o f c ′ on C . The path C ( b, c ) has even le ngth, other wise ( aa ′ , bb ′ , cc ′ ) would be a go o d triple and ( C, C ′ ) a go o d pair , a contradiction. It follows tha t the vertex c has a neighbo r, say d on C ( b, c ) and C ( b, d ) has o dd length. Let d ′ be the neig hbor of d on C ′ . It must be p ointed out that d ′ is a vertex of C ′ ( a ′ , b ′ ). As a matter o f fact if on the contrary d ′ belo ngs to C ′ ( c ′ , a ′ ) we 9 would hav e a goo d triple with ( dd ′ , cc ′ , bb ′ ) when C ′ ( c ′ , d ′ ) has odd length and with ( aa ′ , bb ′ , dd ′ ) when C ′ ( c ′ , d ′ ) is a n ev en path; a contradiction in b oth cases . But now by the choice of the edge ab the leng th of C ′ ( a ′ , b ′ ) cannot b e g reater than C ′ ( d ′ , c ′ ), thus d ′ is adjacent to a ′ and the path C ′ ( d ′ , b ′ ) has even length.  W e hav e : Theorem 3.12. L et G b e a p ermutation gr aph then τ ( G ) ≤ 4 or G is the Petersen gr aph. Pro of L e t C a nd C ′ the 2 − factor of chordless cycles which par tition V ( G ) a nd W e can assume that G is not 3 − edge colo urable o ther wise τ ( G ) = 3 and there is nothing to prove. Hence, C and C ′ hav e b oth o dd lengths. In addition we assume that ( C, C ′ ) is not a g o o d pair, otherwise we are done by Theorem 3.5. Let x 1 x 2 be an edge of C s uch that the o dd pa th deter mined o n C ′ with the neighbors of x 1 and x 2 , say y 1 and y 2 resp ectively , has minimum length. W e cho ose to or ie n t C from x 1 to x 2 and to or ient C ′ from y 1 to y 2 . Th us C ( x 1 , x 2 ) is a n edge and C ′ ( y 1 , y 2 ) is a n odd path. By Lemma 3.11 we must hav e tw o vertices x 3 and x 4 on C and their neighbor s y 3 and y 4 on C ′ such that C ( x 4 , x 3 ), C ′ ( y 1 , y 4 ), C ′ ( y 2 , y ′ 3 ) ar e edges, C ( x 2 , x 4 ) b eing an o dd path while C ′ ( y 4 , y 2 ) has even length. Claim 1. The vertic es y 1 and y 3 ar e adjac ent . Pro of Ass ume not. The o dd path C ′ ( y 4 , y 3 ) having the sa me leng th than C ′ ( y 1 , y 2 ) we may ap- ply Lemma 3.11 o n the edge x 4 x 3 ( x 4 = a , x 3 = b ). Th us ther e is edges; sa y x 5 y 5 and x 6 y 6 , x 5 and x 6 being vertices of C , y 5 and y 6 vertices o f C ′ , the paths C ( x 6 , x 5 ), C ′ ( y 4 , y 6 ) and C ′ ( y 3 , y 5 ) having length 1. More over the paths C ( x 3 , x 6 ) and C ′ ( y 6 , y 2 ) are o dd. Since it is assumed that y 1 and y 3 are indep endent we hav e y 5 6 = y 1 and x 5 6 = x 1 . Observe that the pa ths C ′ ( y 1 , y 2 ) and C ′ ( y 6 , y 5 ) have the same length, thus we apply Lemma 3.11 a gain with a = x 6 and b = x 5 . Let y 7 be the neigh b or of y 5 on C ′ y 5 , y 1 ) and x 7 be the neigh b or of y 7 on C . W e know that x 7 is a vertex of C ( x 5 , x 1 ) at even distance o f x 5 . The vertex x 8 being the neighbor of x 7 on C ( x 5 , x 7 ) and y 8 the neighbor of x 8 on C ′ , we hav e that y 8 is the neighbor o f y 6 on C ′ ( y 6 , y 2 ). The path C ′ ( y 8 , y 2 ) has even length, hence ther e must be on this path a neighbor of y 8 distinct from y 2 , say y 9 . Let x 9 be the neighbo r of y 9 on C . The vertex x 9 belo ngs to C ( x 7 , x 1 ). Otherwise when x 9 is o n C ( x 2 , x 4 ); if the path C ( x 2 , x 9 ) is o dd we can find a go o d triple, namely ( x 8 y 8 , x 9 y 9 , x 2 y 2 ) on the other case we hav e the go o d triple ( x 9 y 9 , x 4 y 4 , x 1 y 1 ). A c ontradiction in both cases . W e g e t a similar contradiction if x 9 belo ngs to C ( x 3 , x 6 ) by consider ing the triples ( x 5 y 5 , x 9 y 9 , x 8 y 8 ) or ( x 9 y 9 , x 4 y 4 , x 2 y 2 ). Finally , when x 9 is a vertex of C ( x 5 , x 8 ) a contradiction o ccurs with the tr iple ( x 5 y 5 , x 9 y 9 , x 7 , y 7 ) if C ( x 5 , x 9 ) is o dd and with the tr ip e ( x 8 y 8 , x 9 y 9 , x 6 y 6 ) other- wise. 10 J.L. FOUQUET AND J.M. V ANHERPE y 1 odd x 9 x 8 x 7 x 6 x 5 1 1 odd odd y 9 y 8 y 6 odd 1 1 1 y 7 y 5 1 1 odd C ′ C y 4 y 3 y 2 x 3 x 4 1 1 1 1 x 2 x 1 Figure 7 . Situation a t the end of C la im 2 Observe that the path C ( x 7 , x 9 ) must b e o dd or ( x 9 y 9 , x 7 y 7 , x 8 y 8 ) would b e a go o d triple, a contradiction. But now ( x 9 y 9 , x 5 y 5 , x 4 y 4 ) is a go o d triple, a contradiction which pr oves the Claim (see Figur e 7).  F r om now on w e assume that y 3 y 1 is an edge. The path C ( x 1 , x 3 ) b eing o dd there must b e a neighbo r of x 3 on C ( x 3 , x 1 ) distinct fr o m x 1 , le t x 5 be this vertex. It’s neighbor on C ′ , say y 5 , must b e on C ′ ( y 4 , y 2 ). Moreover the length of C ′ ( y 4 , y 5 ) is o dd otherwise the edges x 5 y 5 , x 3 y 3 and x 1 y 1 would form a go o d triple, a co n tradiction. Claim 2. The p aths C ′ ( y 4 , y 5 ) and C ′ ( y 5 , y 2 ) ar e r e duc e d to e dges. Pro of Assume in a first stage that the neig h bo r of y 4 on C ′ ( y 4 , y 5 ) is distinct from y 5 , let y 6 be this vertex and x 6 be its neighbo r on C . The vertex x 6 cannot b elong to C ( x 5 , x 1 ), otherwise we would hav e a go o d triple ( x 3 y 3 , x 6 y 6 , x 4 y 4 ) when C ( x 5 , x 6 ) is an even path and the go o d triple ( x 4 y 4 , x 6 y 6 , x 2 y 2 ) if it’s an o dd path, contradictions. Similarly the vertex x 6 cannot b elong to C ( x 2 , x 4 ). On the contrary we would hav e a go o d triple with the edges x 2 y 2 , x 6 y 6 and x 1 y 1 when the path C ( x 2 , x 6 ) is o dd and another go o d triple with the edg e s x 4 y 4 , x 6 y 6 and x 1 y 1 . On the s a me manner we can prov e that the path C ′ ( y 5 , y 2 ) has length 1.  It co mes fr o m Claim 2 tha t C ′ has only 5 vertices. Since b oth cycles C and C ′ hav e the same length C has 5 vertices too and G is the Petersen graph.  In [16] W atkins prop osed two fa milies of genera lized Blanu ˇ sa snarks using the blo cks B , A 1 and A 2 describ ed in Figure 8. The genera lized Bla nu ˇ sa snarks of type 1 (r esp. of type 2) ar e o btained by consider ing a num b er o f blo cks B and one blo ck A 1 (resp. A 2 ), these blo cks ar e arr a nged cyclic a lly , the semi-edg es a a nd b of o ne blo ck b eing connected to the semi-edge s a , b of the next one. Recently generalized Blanu ˇ sa snarks w ere studied in ter ms of circular chromatic index (see [9, 7]). The gener a lized Blan u ˇ sa sna rks are p ermutation graphs, hence : Corollary 3. 13. L et G b e a gener alize d Blanu ˇ sa snarks then τ ( G ) = 4 . 11 a b ′ a ′ b a b ′ a ′ b Block A 2 Block A 1 Block B b a a ′ b ′ Figure 8 . Blo cks for the constructio n of generalized Blanu ˇ sa snarks. 4. On graphs with τ ≥ 5 It is an easy task to cons truct cubic graphs with pe r fect matching index at le ast 5 with the help of P rop osition 2 .2. T ake indeed the Petersen g r aph P and any bridgeless cubic gr aph G and apply the c o nstruction P J G . Prop ositio n 4.1. L et G b e bridgeless cubic gr aph with p erfe ct matching index at le ast 5 and let H b e a c onne cte d bip artite cubic gr aph. Then G ⊗ H is bridgeless cubic gr aph with p erfe ct matching index at le ast 5 . Pro of Ass ume that τ ( G ⊗ H ) = 4 and let M = { M 1 , M 2 , M 3 , M 4 } be a covering of its edge s et into 4 p erfect matchings. Le t { aa ′ , bb ′ , cc ′ } (with a, b and c in G and a ′ , b ′ and c ′ in H ) be the principal 3 − edge cut of G ⊗ H . F rom Item 2 of Prop ositio n 2.3 there is p erfect matching M i ∈ M such that { aa ′ , bb ′ , cc ′ } ⊆ M i . This is clearly imp oss ible since the set of vertices of H which must b e saturated by M i is partitio ne d into 2 indep endent sets whose size differs by one unit.  Let us consider the following co nstruction. Giv en four cubic gra phs G x 1 1 , G x 2 2 , G x 3 3 , G x 4 4 together with a distinguished vertex x i ( i = 1 , 2 , 3 , 4) whose ne ig hbors in G x i i are a i , b i and c i , we get a 3-connected cubic gr aphs in deleting the vertices x i ( i = 1 , 2 , 3 , 4) and co nnecting the remaining subg raphs as de s crib ed in Figure 9. In other words we define the cubic graphs denoted K 4 [ G x 1 1 , G 2 x 2 , G x 3 3 , G x 4 4 ] who se vertex set is [ i ∈{ 1 , 2 , 3 , 4 } V ( G x i i ) − [ i ∈{ 1 , 2 , 3 , 4 } { x i } while the edge set is [ i ∈{ 1 , 2 , 3 , 4 } E ( G x i i ) − [ i ∈{ 1 , 2 , 3 , 4 } { a i x i , b i x i , c i x i } [ { a 1 c 3 , b 1 a 4 , c 1 c 2 , b 2 c 4 , a 2 c 3 , b 3 b 4 } . F o r con venience G i ( i ∈ { 1 , 2 , 3 , 4 } ) will denote the induced subgraph of G x i i where the vertex x i has b een deleted. Prop ositio n 4.2. L et G x 1 1 , G x 2 2 , G x 3 3 and G x 4 4 b e 3 -c onne cte d cubic gr aphs such that τ ( G x 1 1 ) ≥ 5 , τ ( G x 2 2 ) ≥ 5 , G 4 is r e duc e d t o a single vertex, say x . Then τ ( K 4 [ G x 1 1 , G x 2 2 , G x 3 3 , G y 4 ]) ≥ 5 . Pro of Let us denote G = K 4 [ G x 1 1 , G x 2 2 , G x 3 3 , G x 4 4 ]. Obser ve that a 4 = b 4 = c 4 = x . If τ ( G ) = 3 the g raph G would b e 3- e dge coloura ble, but in cons idering the 3-edge cut { a 1 a 3 , b 1 a 4 , c 1 c 2 } we would have χ ′ ( G x 1 1 ) = 3, a co n tradiction. Hence 12 J.L. FOUQUET AND J.M. V ANHERPE G 1 G 4 b 4 c 4 a 4 a 1 a 1 c 2 b 2 a 2 b 3 c 3 c 1 b 1 G 3 G 2 Figure 9 . K 4 [ G x 1 1 , G x 2 2 , G x 3 3 , G x 4 4 ] τ ( G ) ≥ 4. Assume that τ ( G ) = 4 and let M = { M 1 , M 2 , M 3 , M 4 } b e a covering of its edge set into 4 p erfect matchings. F r om Item 2 of P rop osition 2 .3 there is pe rfect matching M i ∈ M such that { a 1 a 3 , b 1 a 4 , c 1 c 2 } ⊆ M i . F o r the same rea son, there is p erfect ma tc hing M j ∈ M such tha t { c 1 c 2 , xb 2 c 3 a 2 } ⊆ M j . W e cer tainly hav e i 6 = j , o therwise the vertex x is inciden t twice to the same p erfect ma tc hing M i . Without loss of generality , we suppo se that i = 1 and j = 2. Hence c 1 c 2 ∈ M 1 ∩ M 2 . If we consider the 3 − edge cut { a 1 a 3 , b 1 a 4 , c 1 c 2 } , s ince ea ch p erfect ma tc hing must intersect this c ut in an o dd nu mber of edges we m ust hav e one of the edges a 1 a 3 or b 1 x in M 3 while the other m ust b e in M 4 . T he same holds with the 3 − edge cut { c 1 c 2 , xb 2 c 3 a 2 } and the edges b 2 x and a 2 c 3 . Hence, we c an supp ose that a 1 a 3 ∈ M 1 ∩ M 3 and b 1 x ∈ M 1 ∩ M 4 as well that b 2 x ∈ M 2 ∩ M 3 and a 2 c 3 ∈ M 2 ∩ M 4 , a co n tradiction s ince the s et of edges co nt ained into 2 p erfect matchings of M is a p erfect matching by Item 2 of Prop ositio n 3.1 a nd x is incident to tw o such edges.  W e do not kno w any cyc lic ally 4 − edge connected c ubic graph, distinct from the Petersen gr aph, having a p erfect matching index a t lea s t 5 and we pro po se as an op en problem: Problem 4.3. Is t her e any cyclic al ly 4 − e dge c onne cte d cubic gr aph distinct fr om the Petersen gr aph with a p erfe ct matching index at le ast 5 ? 5. Technical tools. In fa c t Theo rem 3.5 can b e g e neralized. Let M b e a p erfect matching, a s e t A ⊆ E ( G ) is a M − b alanc e d matching w he n we can find a p erfect matchings M ′ such that A = M ∩ M ′ . Assume that M = { A, B , C } are 3 pair wise disjoint 13 M − balanced matchings, we s hall say that M is a go o d family whenever the tw o following co nditions are fulfilled: i Every o dd cycle C of G \ M ha s ex actly one vertex incident with one edge of each s ubs e t of M and the three paths de ter mined by thes e vertices on C a re o dd. ii F or every even c ycle of G \ M there a re at least t wo matchings o f M with no edge incident to the cycle. Theorem 5.1. L et G b e a bridgeless cubic gr aph to gether with a go o d family M . Then τ ( G ) ≤ 4 . Sk etch o f the proof L e t us denote M A (resp. M B , M C ) a p erfect matching such that M A ∩ M = A (resp. M B ∩ M = B , M C ∩ M = C ). Let C b e a c ycle of the 2 -factor G − M . When C is an even cycle , there are precisely tw o matchings on C , namely M C and M ′ C such that M C ∪ M ′ C cov er s all the edge-s et of C . Since there ar e at least tw o matchings in { M A , M B , M C } that ar e no t incident to C , say M A and M B , up to a redistribution of the edges in M A ∩ C and M B ∩ C we ma y a ssume that M C ⊂ M A and M ′ C ⊂ M B . If C is an o dd cycle we know that C has precisely one vertex which is incident to A sa y a , one vertex which is inciden t to B say b , one vertex which is incident to C say c . Without loss of generality w e may assume tha t there is an or ientation of C such that the pa th C ( a, b ) ha s o dd length a nd the vertex c in C ( b, a ). W e k now that the path C ( b, c ) is o dd th us the edge-set of C is covered with M A ∪ M B ∪ M C .  In the same ma nner we can obtain a theorem insur ing the ex istence of a 5- cov er ing . Assume that M = { A, B , C, D } are 4 pairwise disjoint M − balanced matchings, we shall say that M is a ic e family whenever the tw o following conditions are fulfilled: i Every o dd cycle C of G \ M ha s ex actly one vertex incident with one edge of each subset of M and at le ast tw o disjoint paths determined by these vertices on C ar e o dd. ii F or every even c y cle of G \ M there a re at leas t t wo matchings of M with no edge incident to the cycle. Theorem 5. 2. L et G b e a bridgeless cu bic gr aph to gether with a nic e family M . Then τ ( G ) ≤ 5 . Pro of Let us denote M A (resp. M B , M C , M D ) a p erfect matching such that M A ∩ M = A (resp. M B ∩ M = B , M C ∩ M = C , M D ∩ M = D ). Let C b e a c ycle of the 2 -factor G − M . When C is an even cycle , there is at least tw o matchin gs in { M A , M B , M C , M D } that ar e no t incident to C , sa y M 1 and M 2 . As in Theo rem 5.1 we may assume that the edge-se t of C is a subset of M 1 ∪ M 2 . If C is an odd cycle w e kno w that C has precisely one vertex which is incident to A say a , one vertex which is inc ide nt to B say b , one v er tex which is incident to C say c , o ne vertex which is incident to D say d . Without lo s s of genera lit y we may assume that there is an orientation of C s uch that the path C ( a, b ) has o dd length and the vertices c and d are in this order in ( b , a ). W e can supp ose that the path 14 J.L. FOUQUET AND J.M. V ANHERPE ( b, c ) is even otherwis e the edge-set of C would b e cov er ed with M A ∪ M B ∪ M C . But now, since C is an o dd cycle the pa th C ( d, a ) has o dd leng th and the e dge-set of C is a s ubs e t of M A ∪ M B ∪ M D and ( M , M A , M B , M C , M D ) is a 5 -cov er ing.  In a for thcoming pap er [5] we shall give a n analog o us theorem insur ing the existence of a F ulkerson cov er ing a nd some applicatio ns. 6. Odd or even cove rings. A covering o f a bridge le ss cubic graph b eing a set of p erfect matchings s uch that every edge is con tained in at least one p erfect matching, w e define an o dd c overing as a co vering such that eac h edge is contained in an o dd num ber of the mem b er s of the covering. In the sa me way , a n even c overing is a covering suc h that each edge is contained in a n even num b er (at least 2) members of the cov e ring. The s ize o f an o dd (or even) covering is its num b er of members . As so on a s a cov er ing is given an even c ov ering is obtained by taking each perfect ma tc hing twice. Prop ositio n 6.1. L et G b e bridgeless cubic gr aph s u ch that τ ( G ) = 4 . Then G has an o dd c overing of size 5 . Pro of Let G be a cubic gra ph such that τ ( G ) = 4 and let M = { M 1 , M 2 , M 3 , M 4 } be a covering of its edge s e t in to 4 perfect matchings. L e t M b e the p erfect matc h- ing formed with the edge s contained in exac tly tw o p erfect matchings o f M . Then we can chec k that { M , M 1 , M 2 , M 3 , M 4 } cov er every edge o f G either one time or three times.  Prop ositio n 6. 2. L et G b e a bridgeless cubic gr aph to gether with an o dd c overing M of size k . Then either G has an o dd c overing of size k − 2 or ∀ M , M ′ ∈ M we have M 6 = M ′ . Pro of Assume that there ar e tw o ident ical p erfect matchings M a nd M ′ in M . Each edge e covered by M (and thus M ′ ) must b e cov er ed by at leas t a nother per fect matc hing M e and the set M − { M , M ′ } is still an o dd cov er ing. The result follows.  Prop ositio n 6.3. The Petersen gr aph has no o dd c overing. Pro of Let M be a n odd cov er ing of the Petersen graph with minimum s iz e . Then, by Pro po sition 6.2 M must b e a set of distinct p er fect ma tc hings. The Petersen graph has exactly 6 distinct perfect matc hings (inducing a F ulkerson covering, that is an even cov er ing) and it is an ea s y task to chec k that any subset of 5 p erfect matchings is not an o dd cov ering . Since τ ( P etersen ) = 5, the r e s ult follows.  Seymour ([12]) remarked tha t the edge set of the Petersen graph is no t express ible as a symmetric difference (mo d 2) of its p er fects matchings. Problem 6. 4. Which bridge less cubic gr aph c an b e pr ovide d with an o dd c overing ? W e remar k that 3 − edge-color able cubic graphs a s well as bridgeless cubic graph with perfect matching index 4 hav e an o dd cov ering (with size 3 and 5 resp ectively). Prop ositio n 6 .5. L et G b e bridgeless cubic gr aph without any o dd c overing and let H b e a c onn e cte d bip artite cubic gr aph. Then G ⊗ H has no o dd c overing. 15 Pro of Assume that G ⊗ H ca n b e provided with an o dd cov ering M . Let { aa ′ , bb ′ , cc ′ } (with a, b and c in G and a ′ , b ′ and c ′ in H ) b e the principa l 3 − edge cut of G ⊗ H . None of the p erfect matchings of M can contain the principal 3 − e dg e cut since the s et of vertices of H which must b e s aturated by such a p erfect matc h- ing is pa r titioned into 2 indep endent sets who se size differs by one unit. Hence every p erfect matching of M ∈ M contains e x actly one edge in { aa ′ , bb ′ , cc ′ } a nd leads to a p erfect matc hing M ′ of G . The set M ′ of perfect matchings so obtained is an o dd covering of G , a contradiction.  Prop ositio n 6.6. L et G x 1 1 and G x 2 2 b e cubic gr aphs with distinguishe d vertic es x 1 and x 2 such that τ ( G x i i ) ≥ 5 ( i = 1 , 2 ) and τ odd ( G x i i ) 6 = 5 ( i = 1 , 2 ) . L et G x ′ 4 and G y ′ 3 b e t wo c opies of the cubic gr aph on two vertic es and G = K 4 [ G x 1 1 , G x 2 2 , G y ′ 3 , G x ′ 4 ] , then τ ( G ) ≥ 5 and if τ odd G is define d then τ odd ( G ) 6 = 5 . Pro of Let x and y be resp ectively the unique vertex of G 4 , G 3 (see Figure 9 where G 4 is r educed to a single vertex x and G 3 is r educed to y ). W e know b y Pro po sition 4.2 that τ ( G ) ≥ 5. Assume tha t τ odd ( G ) = 5 and let M = { M 1 , M 2 , M 3 , M 4 , M 5 } be an o dd 5- cov er ing. The p erfect ma tc hings of M a re pairwise distinct otherw is e by Pro po sition 6 .2 either G x 1 1 or G x 2 2 would b e 3 -edge colo r able, a contradiction. Observe tha t ea ch vertex is incident to one edge tha t b elong s to pre cisely three matchings o f M , the tw o o ther edges b eing cov er ed o nly once. Mor eov er , the set of edges that b elong to 3 matchings of M is a p erfect ma tc hing itself. The 3 -edge cut { a 1 y , b 1 x, c 1 c 2 } m ust b e entirely contained in s ome matching o f M , say M i otherwise we would hav e a 5- o dd covering of G x 1 1 , a contradiction. Similarly there is a per fect matching in M , say M j that contains the edges c 1 c 2 , b 2 x , a 2 y . Thus the edg e c 1 c 2 m ust b elong to 3 ma tchings of M . Without loss of generality w e ass ume that i = 1, j = 2 and c 1 c 2 ∈ M 1 ∩ M 2 ∩ M 3 . If y a 1 ∈ M 3 , s ince a p erfect matching in tersects a n y o dd cut in a n o dd nu mber of edges we hav e xb 1 ∈ M 3 , it fo llows that the e dge y a 1 m ust b e a member of a third matching o f M as well as the edge xb 1 . If for s ome k we have y a 1 ∈ M k and xb 1 ∈ M k , k ∈ { 2 , 4 , 5 } , k b eing o bviously distinct from 2 M k int ersects the 3-edge c ut in an even num b er of edges, a co nt radiction. Hence we may assume that y a 1 ∈ M 4 and xb 1 ∈ M 5 . B ut now the edge xy is cov ered b y none of the matchings of M , a contradiction. Consequently y a 1 / ∈ M 3 , similarly xb 1 / ∈ M 3 . If ya 1 ∈ M 4 this edge m ust b e lo ng to a thir d matc hing of M w hich is M 5 . Since the set of edges that are covered 3 times is a p erfect matching xb 1 ∈ M 4 ∩ M 5 . B ut in this case the e dge c 1 c 2 would b elong to M 4 and M 5 , a contradiction. It follows that y a 1 as w ell as xb 2 are cov er e d only once and the edge xy be lo ngs to 3 matchings of M , that is xy ∈ M 3 ∩ M 4 ∩ M 5 . But now, neither M 4 nor M 5 int ersect the edge-cut { y a 1 , xb 1 , c 1 c 2 } a contradiction since a p erfect matching must int ersect every odd e dge-cut in an o dd num be r of edges.  The g raph G depic ted in Figure 10 is an example of cubic gr aphs with a 7-o dd cov er ing a nd a p erfect matching index equa ls to 5. W e know by P rop osition 6.6 16 J.L. FOUQUET AND J.M. V ANHERPE 10 8 9 5 4 14 15 19 18 16 17 7 6 11 2 12 13 3 1 0 Figure 1 0. A gr aph G such that τ ( G ) = 5 and τ odd ( G ) = 7. that τ odd ( G ) ≥ 7 . As a matter of fact, this graph has 20 distinct p erfect matchin gs and among all the 7-tuples o f p erfect matchings (7752 0) 64 fo r m an o dd-covering. Let us give below such a 7 -tuple. { 0 − 10 , 1 − 5 , 2 − 9 , 3 − 13 , 4 − 8 , 6 − 7 , 11 − 15 , 12 − 19 , 14 − 18 , 16 − 17 } { 0 − 1 , 2 − 8 , 3 − 4 , 5 − 9 , 6 − 7 , 10 − 12 , 11 − 15 , 13 − 14 , 16 − 18 , 17 − 19 } { 0 − 1 , 2 − 10 , 3 − 13 , 4 − 5 , 6 − 8 , 7 − 9 , 11 − 15 , 1 2 − 19 , 1 4 − 18 , 16 − 1 7 } { 0 − 1 , 2 − 10 , 3 − 13 , 4 − 8 , 5 − 9 , 6 − 7 , 11 − 16 , 1 2 − 18 , 1 4 − 15 , 17 − 1 9 } { 0 − 11 , 1 − 5 , 2 − 9 , 3 − 13 , 4 − 8 , 6 − 7 , 10 − 12 , 14 − 15 , 16 − 18 , 17 − 19 } { 0 − 11 , 1 − 5 , 2 − 9 , 3 − 13 , 4 − 8 , 6 − 7 , 10 − 12 , 14 − 18 , 15 − 19 , 16 − 17 } { 0 − 11 , 1 − 6 , 2 − 1 0 , 3 − 7 , 4 − 8 , 5 − 9 , 12 − 19 , 13 − 17 , 14 − 15 , 16 − 18 } Moreov er the following p erfect matchings form a 5-covering. { 0 − 1 , 2 − 10 , 3 − 13 , 6 − 8 , 7 − 9 , 4 − 5 , 12 − 19 , 1 6 − 17 , 1 4 − 18 , 11 − 1 5 } { 2 − 9 , 1 − 6 , 7 − 9 , 4 − 5 , 3 − 13 , 0 − 11 , 10 − 12 , 14 − 15 , 16 − 18 , 17 − 19 } { 1 − 6 , 7 − 9 , 2 − 8 , 5 − 4 , 0 − 10 , 1 2 − 18 , 17 − 1 9 , 14 − 15 , 1 1 − 15 , 3 − 13 } { 0 − 1 , 2 − 8 , 6 − 7 , 5 − 9 , 3 − 4 , 10 − 12 , 13 − 17 , 14 − 18 , 15 − 19 , 11 − 16 } { 1 − 6 , 5 − 9 , 4 − 8 , 3 − 7 , 2 − 10 , 0 − 11 , 12 − 18 , 13 − 14 , 15 − 19 , 16 − 17 } 17 W e do not know any ex ample of graph G for which τ odd is defined and with τ ( G ) = τ odd ( G ) = 5. W e just obser ve that in such a gr a ph every vertex would b e incident to an edge b elonging to 3 p erfect matchings a nd to prec is ely tw o edges cov er e d only once. The set o f edges cov ered by 3 perfect matchings b eing a p erfect matching its e lf. Problem 6.7. Is it true that every bridgeless cu bic gr aph has an even c overing wher e e ach e dge app e ars twic e or 4 t imes ? The a nswer is yes for 3 − edg e -colora ble cubic gr aphs and for bridgeless cubic graphs with per fect matchin g index 4 since such graphs hav e an even cov er ing of size 8. References 1. D. Bl an u ˇ sa, Pr oblem c eterij u b or a (The pr oblem of four c olors) , Hrv atsko Pr iro doslov ono Dru ˇ stvo Glasnik M at-Fiz. As tr. 1 (1946), 31–42. 2. M. Ell ingham, Petersen sub divisi ons in some r egul ar gr aphs , Congre. Numer. 44 (1984), 33– 40. 3. G. F an and A. Raspaud, F ulkerson ’s c onje ctur e and cir cuit c overs , J. Com b. Theory Ser. B 61 (1994), 133–138. 4. J.L. F ouquet and J.M. V anherp e, On a c onje ctur e fr om K aiser and Rasp aud on cubic g r aphs , T ech. rep ort, LIFO, september 2008. 5. , O n Fulkerson c onje ct ur e , T ech. rep or t, LIFO, 2009. 6. D.R. F ulkerson, Blo cking and anti-blo cki ng p airs of p olyhe dr a , Math. Programmi ng 1 (1971), no. 69, 168–194. 7. M. Ghebleh, Cir c ular Chr omatic Index of Gener alize d Blanu ˇ sa Snarks. , The Electronic Jour- nal of Comb inatorics (2008). 8. R. Isaacs, Infinite families of non-trivial triv alent gr aphs which ar e not T ait c olor able , Am. Math. Month ly 82 (1975), 221–239. 9. J. Maz´ ak, Cir cular chr omatic index of typ e 1 Blanu ˇ sa snarks , J. Gr aph Theory 59 (2008), no. 2, 89–96. 10. T. Sch¨ onberger, Ein Beweis des Peterschen Gr aphensatzes , Acta Sci. Math. Szeged 7 (1934), 5157. 11. P . D. Seymour, Gr aph the ory and r elate d topics , pp. 342–3 55, J.A. Bondy and U.S.R. Murty , eds., A cademic Press, 1979. 12. , On Multi- Colourings of Cubic gr aphs, and Conjectur es of F ulkerson and Tutte , Pro- ceedings of the London Mathematical So ci ety 38 (1979), no. 3, 423–460. 13. G. Szek er` es, Polyhe dr al de c omp ositi ons of cubic gr aphs , Bull. Austral. Math. Soc. 8 (1973), 367–387. 14. S. Norine T. Kaiser, D. Kr´ al, Unions of p erfe c t matchings in cubic gr aphs , Electronic Notes in Di screte Mathematics 2 2 (2005), 341–345. 15. W.T. T utte, A c ontrib ut i on on the the ory of chr omatic p olynomial , Canad. J. Math 6 (19 54), 80–91. 16. J. J. W atkins, Gr aph the ory and its applic ations , c h. Snarks, New Y ork Acad. Sci., 1989. 17. R. X.Hao, J. B Niu, X . F. W ang, C. Q Zhang, and T. Y. Zhang, A note on b er gefulkerson c oloring , Discrete Mathematics In Press (2009). 18. C-Q Zhang, Inte ger flows and cycle c overs of gra phs , Pure and A pplied Mathematics, Dekk er, 1997. L.I.F.O., F acul t ´ e des Sciences, B.P. 6759 Universit ´ e d’Orl ´ eans. 45067 Orl ´ eans Cedex 2, FR