Non-separating 2-factors of an even-regular graph

Transcription

1 Discrete Mathematics ) Non-separating -factors of an even-regular graph Yusuke Higuchi a Yui Nomura b a Mathematics Laboratories College of Arts and Sciences Showa University 456 Kamiyoshida Fuiyoshida Yamanashi Japan b Department of Mathematics Tokyo Institute of Technology Oh-okayama Meguro-ku Tokyo Japan Received January 007; received in revised form 9 August 007; accepted 3 October 007 Available online 6 November 007 Dedicated to Professor Hikoe Enomoto on his sixtieth birthday. Abstract For a -factor F of a connected graph G we consider G F which is the graph obtained from G by removing all the edges of F. If G F is connected F is said to be a non-separating -factor. In this paper we study a sufficient condition for a r-regular connected graph G to have such a -factor. As a result we show that a r-regular connected graph G has a non-separating -factor whenever the number of vertices of G does not exceed r + r. c 007 Elsevier B.V. All rights reserved. Keywords: -factor; Non-separating; Even-regular graph. Introduction Let G = V G) EG)) be a graph with set of vertices V G) and set of edges EG). All graphs in this paper are finite and simple. We sometimes write simply G for the number of vertices V G). The neighbourhood of x V G) is denoted by N G x) and the degree of x by d G x) = N G x). If d G x) is a constant positive integer k independent of the choice of x V G) G is called a regular graph or k-regular; moreover if d G x) is a constant positive even integer G is called even-regular. A -factor of G is a -regular spanning subgraph of G. For a subgraph H of G the graph such that the set of vertices is V G) and that of edges is EG) \ EH) is also a subgraph of G. We write simply G H for this resultant graph. The subgraph of G induced by a set of vertices S V G) is denoted by S G. We let N G S) = {y V G) \ S x S and xy EG)} denote the neighbourhood of a set of vertices S in G. Terminology and notation not defined here can be found in []. There exist many kinds of sufficient conditions for a graph to have a -factor. The following is a famous and classical result: Theorem Petersen [3]). A graph G is -factorable if and only if G is r-regular for some positive integer r. Corresponding author. addresses: higuchi@cas.showa-u.ac.p Yu. Higuchi) nomura@math.titech.ac.p Y. Nomura) X/$ - see front matter c 007 Elsevier B.V. All rights reserved. doi:0.06/.disc

2 Yu. Higuchi Y. Nomura / Discrete Mathematics ) Fig... The r-regular graph G having no NSF with G = f r) +. Here G is called -factorable if a family of edge-disoint -factors of G covers the set of edges of G. In general a connected and even-regular graph G has many types of -factors but we do not know whether there exists a -factor of G such that G F is connected; that is in general it holds that ωg F) where ωh) is the number of components of H. In this paper we study a sufficient condition for an even-regular connected graph G to have a -factor F satisfying G F is connected; we call such a -factor of G a Non-Separating -Factor NSF). Let r be an integer with r and G a r-regular graph. Our obect is to determine a function f on r which is optimal in some sense satisfying the following: a r-regular connected graph G has a NSF if G f r). We immediately have f r) = 4r as such a function f by combining Petersen s theorem and also a classical and famous result by Dirac []: a graph G has a Hamilton cycle if d G x) G / for every x V G). Of course since G 4r = d G x) for any x V G) G has a Hamilton cycle or equivalently a connected -factor C. We consider G C as a r )-regular subgraph of G then G C has a -factor F. This F becomes a NSF since G F has the connected -factor C. Another function f r) = r + 4 was suggested in [6]. Let us sketch its proof. If r = that is f ) = 8 then every 4-regular connected graph G with at most 8 vertices has a Hamilton cycle; thus G has a NSF. Let us assume that every k-regular connected graph G with G f k) has a NSF for every k = 3... r. Suppose ωg F) for a -factor F of a r-regular connected graph G with G f r). Then every component M i of G F is r )-regular M i r and it holds that M i f r) r ) = f r ). By induction there exists a NSF say F i of M i for every i. Putting F = i F i we can see F is the desired NSF of G. We do not believe the above f is optimal but it was pointed out in [7] that the optimal upper bound f r) = Or ). Our main theorem in this paper is as follows: Theorem. Let G be a r-regular connected graph where r is an integer with r. If G f r) then G has a non-separating -factor NSF) where f r) = r + r. The function given above is optimal in the following sense: Let K r+ be the graph obtained from the complete graph K r+ with r + ) vertices by removing an edge e. Let M i be graph-isomorphic to K r+ for i =... r; we let x i and y i denote the non-adacent vertices in M i. We set a graph G Fig..) as V G) = {v} r i= V M i)) and EG) = {vx i vy i i =... r} r i= EM i)). Then for every r G is r-regular G = f r) + = r + r + and G has no NSF. This work is a by-product obtained in our research on the relationship between the covering structure and the properties of spectra of the discrete Laplacians [4].. Preliminary We give the proof of Theorem by induction on r. Let us put r = at the first stage of induction. Recall f r) = r + r hence f ) = 0. Lemma.. Every 4-regular connected graph G with G 0 has a NSF. To prove the above we use the following result by Bill Jackson. Theorem. Jackson [5]). Let G be a -connected k-regular graph with G 3k. Then G has a Hamilton cycle.

3 5540 Yu. Higuchi Y. Nomura / Discrete Mathematics ) Proof of Lemma.. It is easy to check that every 4-regular connected graph with at most 0 vertices is -connected. In fact if there exists a vertex x 0 V G) such that ωg x 0 ) then we can find a component M such that M 4 where G x 0 = V G) \ {x 0 } G. Here for any x V M) it holds that 3 d M x) M. Thus it must hold that M = 4 and that d M x) = 3. This implies that N G x 0 ) = V M) since G is 4-regular which contradicts the fact that x 0 is a cut vertex of G. Then Theorem. yields that G has a Hamilton cycle H. The subgraph G H is -regular that is it is a -factor of G and moreover it is a NSF of G. Now let us set the following hypothesis for a fixed r 3: Hypothesis.3. For every k = 3... r every k-regular connected graph G with G f k) has a NSF. Under Hypothesis.3 we only have to show that every r-regular connected graph G with G f r) has a NSF. We will derive a contradiction when assuming that there exists a r-regular connected graph G with G f r) having no NSF. So we suppose the following assumption under Hypothesis.3. Assumption.4. Let G be a r-regular connected graph such that G f r) and G has no NSF. In Section 3 by analyzing the properties of G in the above we show the non-existence of such a graph G. 3. Graph structure under Hypothesis.3 and Assumption.4 First let us give our notation. Notation 3.. For a subgraph F of G let G F = m i= G i where m = ωg F) every G i is a component of G F and G i G for every i < m. Claim 3.. For any -factor F of G it holds that G F = l i= G i such that l 3 and G > f r ) G r for. In particular if l = 3 then we have G = f r ) + G = G3 = Kr and G = f r) = r + r. Proof. For a -factor F we set G F = l i= G i following Notation 3.. By Assumption.4 it must hold that l for any -factor F. Every component G i of G F is r )-regular then G i r for every i. Firstly we show G > f r ). Assume that G f r ). Then G i f r ) for every i and it follows from Hypothesis.3 that G i has a NSF F i for every i. Here we put F = l i= F i then F is a -factor of G; moreover G F is connected. Thus G has a NSF F which contradicts Assumption.4. So it must hold that G > f r ). Secondly we show G f r ). Assume that G > f r ). We have G > f r ) = f r) 4r ) and then G G G < 4r since G f r). So G must satisfy f r ) < G 4r but it contradicts r 3. Thus it is shown that G f r ). Thirdly assume that l 4. From the above we have G > f r ) = f r) 4r ) and G G 3 G 4 r. Then f r) G G + G + G 3 + G 4 f r) 4r ) + ) + 3r ) = f r) + r. This is a contradiction. Finally if l = 3 f r) G = G + G + G 3 f r) 4r ) + ) + G + G 3. Thus we have G + G 3 r ) which implies that G = G 3 = r that is G = G3 = Kr. On the other hand it holds that G f r) r ) = f r ) + thereby G = f r ) +. Claim 3.3. Assume that G has a -factor F such that G F = G G G 3. Let F and F 3 be -factors of G and G 3 respectively. Then for any -factor F of G ωg F F F 3 )) =. Proof. By Claim 3. we know that G = G3 = Kr ; any -factor of K r is a NSF of K r. Then F and F 3 are NSFs of G and G 3 respectively. Moreover for any -factor F of G F = F F F 3 is another -factor of the r-regular connected graph G. By Claim 3. again it holds that ωg F) 3. Assume that ωg F) = 3. Also by Claim 3. G F = N N N 3 where N = G r ) and N = N3 = Kr. Remark that there

4 Yu. Higuchi Y. Nomura / Discrete Mathematics ) exist vertices x V G ) y V G)\V G ) and an edge xy F for = 3. So if V G ) V N k ) for = 3 and k = 3 then V N k ) V G ) which is a contradiction. Therefore we have V N ) V G ) V G 3 ). Let S = V N )\V G ) V G 3 )). By Claim 3. we have N = f r )+ and G + G 3 = r ) f r ). Then S. Remark that N G F F S) =. First we put r = 3. Then S G F F is -regular and S 3. On the other hand S G 4 3 ) f 3) 0 =. This is impossible. Next we put r 4. Then S G F F is r )-regular and S r 3. Moreover S G 4r ) f r) 4r ) < f r ). By Hypothesis.3 every component of S G F F has a NSF which implies that G has a NSF. This is a contradiction. Therefore we obtain that ωg F) 3 so the proof is completed. Now we pursue the following procedure. Procedure Step ). Consider an arbitrary -factor F ) of G and put G F ) = G ) M) where M ) = l = G ) and l = ωg F ) ). According to Notation 3. we have G ) > f r ) G) G) 3 ) r. Now focus on those -factors F ) whose biggest component G ) of G F ) has minimum order. Within this range find and fix a -factor F ) such that l has minimum value. Procedure Step ). Consider an arbitrary -factor F ) of G ) which is r )-regular and put another -factor F ) of G to be F ) = F ) l = F ) ) where F ) is a fixed NSF of G ). Moreover put G F ) = G ) M) where M ) = l = G ) and l = ωg F ) ). Now focus on those -factors F ) such that the biggest component G ) of G F ) has minimum order. Within this range find and fix a -factor F ) such that l has minimum value. Procedure Step k). Pursuing this procedure at Step k we have -factors F ) F )... F k ) and also have M ) M )... M k ) where G F i) = G i) M i) M i) = l i = Gi) and l i = ωg F i) ) for every i =... k. Consider an arbitrary -factor F k) of G k ) = V G) \ k i= V Mi) ) G k i= Fi) which is r k +)-regular. Here we put a -factor F k) of G to be F k) = F k) k i= l i fixed NSF of G i ) where M k) = l k = G k) = Fi) ) where F i) is the for i =... k and F k) is a fixed NSF of G k ). Moreover put G F k) = G k) Mk) and l k = ωg F k) ). Now focus on those -factors F k) such that the biggest component G k) of G F k) has minimum order. Within this range find and fix a -factor F k) such that l k has minimum value. This Procedure defined inductively is terminated at Step r. In the following claim the former statement asserts the existence of the -factors required at Step k + of the Procedure whenever Step k is completed whereas the latter statement follows from the former one together with Claim 3.3 and the construction of the -factors F k). Claim 3.4. V M i) ) V M k) ) = for i < k r. In addition if there exists an integer i such that l i = 3 then l k = for every k i. Proof. First assume that V M ) ) V M ) ). If l = 3 then M ) = 3 = G) and G ) = K r. Moreover is the complete graph with r by the construction of F ) and Claim 3.3 we have M ) = G ). Since G) vertices it must hold that any edge of the -factor F ) incident to a vertex of V G ) ) is always incident to a vertex of V G) \ V G ) ). Then N F )V G ) )) r. Thus in G F ) V G ) ) N G V G ) )) G F ) is connected and moreover V G ) ) N G V G ) )) G F ) r ). Now we may set V G ) ) V G) ) from the assumption that V M ) ) V M ) ) ; then we have G ) r ). Here the fact that V M) ) = r ) and the minimality of G ) imply G) = r ). However the minimality of l contradicts l = 3 > l =. Then l =. Since G ) F ) is connected and N F )V G ) )) we have V M) ) V G ) ) +. This contradicts the minimality of G ). Then we obtain that V M) ) V M ) ) =. Next assume that k r is fixed and that V M i) ) V M ) ) = for any i such that i < k. Here remark that the hypothesis of induction guarantees the existence of the -factor F k+) required at Step k +. Let us show that V M i) ) V M k+) ) = for any i k. If there exists a i k such that V M i) ) V M k+) ) with

5 554 Yu. Higuchi Y. Nomura / Discrete Mathematics ) l i = 3 then by Claim 3.3 and the construction of -factor F k+) at Step k + G F k+) = G k+) G k+) that is l k+ = and M k+) = G k+). We may now assume that V G i) ) V Gk+) ) where G F i) = G i) Mi) M i) = 3 = Gi) and G i) = K r for = 3. By a similar argument as above we have G k+) = r ) and l k+ = which contradicts the choice of the -factor required at Step i. If there exists a i k such that V M i) ) V M k+) ) with l i = then it must hold that V M k+) ) V M i) ) + since M i) F k+) is connected and N F i)v M i) )). This contradicts the minimality of G i) at Step i. Consequently the proof is completed. In addition we also have Claim 3.5. For every i k with i < k N G V M i) )) V M k) ) =. In other words xy EG) for any x V M i) ) and any y V M k) ). Proof. Assume that there exist i k x and y such that xy EG) where i < k x V M i) ) and y V M k) ). We use a similar argument to that in the proof of Claim 3.4 again. First let l i = 3. Then l k = G F i) = G i) 3 = G i) ) where G i) = K r for = 3. As it is seen it holds that N F i)g i) ) r and G i) F k) is connected. The minimality of G i) at Step i implies that V Mk) ) = r ) but l k < l i which contradicts the minimality of l i. Next let l i =. Then we have V M k) ) V M i) ) + according to the connectedness of M i) F k) and the fact that N F i)v M i) )). This contradicts the minimality of G i) at Step i. Definition 3.6. When Step r of the Procedure stated above is done we define the set of vertices V r as V r = V G) \ r i= V Mi) ). For sets of vertices S and T such that S T V G) and S T = we define the set of edges e G S T ) as e G S T ) = {xy EG) x S and y T }. Claim 3.7. V r. e G V r V M i) )) and e G V r V M i) )) F i) for every i r. Moreover for any i < r it holds that e G V r V M i) )) F ) =. Proof. Considering Claims 3.4 and 3.5 together with the construction of F i) we can easily obtain the above. Claim 3.8. l i = for each i =... r. Proof. Suppose l i = 3 for some i. Recall Claim 3.: G F i) = G i) M i) where M i) = 3 = Gi) and = K r for = 3. Suppose V r = and we set {v} = V r. Claim 3.7 says e G V M i) ) V r ) F i) G i) thus there exist z V M i) ) and z V M i) ) such that z z and vz vz EG); it holds that {vz vz } = e G V M i) ) V r ). Since all the graphs G i) and G i) 3 are isomorphic to K r every edge xy E i) oins x G i) and y G i) 3 where Ei) = F i) E V M i) ) G ). Here we have E i) = 4r 3 9. If both z and z are in G i) then we set Ẽi) = EP ) EH ) where P is a Hamilton path in G i) from z to z and H is a Hamilton cycle in G i) 3. Considering F i) = F i) \ E i) ) Ẽ i) we can easily see that F i) is a -factor and that ωg F i) ) = which contradicts the choice of F i) at Step i of the Procedure. The same argument is valid if z and z are in G i) 3. Thus z G i) and z G i) 3. Find and fix an edge w w such that w G i) and w G i) 3. Now we set Ẽ i) = EP ) {w w } EP ) where P is a Hamilton path in G i) from z to w and P is also a Hamilton path in G i) 3 from w to z. Considering F i) = F i) \ E i) ) Ẽ i) we can easily see that F i) is a -factor and that ωg F i) ) = which contradicts the choice of F i) at Step i of the Procedure. Therefore we obtain V r. Now recall that V M i) ) = r ) and V M k) ) r for k i. Then by Claim 3.4 G = V r + V M k) ) + r ) + r )r ) = f r) +. k= This contradicts Assumption.4 thus the proof is completed.

6 Yu. Higuchi Y. Nomura / Discrete Mathematics ) From here on we assume that M k) = G k) for every k =... r. Recall that G F k) = G k) M k) where V M ) ) V M ) ) V M r) ) r. Here V M ) ) V M ) )... V M r) ) and V r are mutually disoint non-empty sets and V G) = V r r k= V M k) ) ). Claim 3.9. If V M k) ) = r for some k then N G V M k) )) V r r; in particular V r r. Furthermore if V M k) ) = r for some k then N G V M k) )) V r r ; in particular V r r. Proof. Let us recall the following: M k) is r )-regular in G F k) e G V M i) ) V M k) )) = for i k from Claim 3.5 and e G V r V M k) )) F k) from Claim 3.7. Assume that V M k) ) = r. Then there exists z x V r such that xz x F k) for every x V M k) ). This implies that e G V r V M k) )) r. Since F k) is a -factor of G we have N G V M k) )) V r r. Next assume that V M k) ) = r ; then M k) becomes the complete graph with r vertices in G F k). There exist two distinct vertices z x and z x such that z x V r and xz x F k) for = and for every x V M k) ); e G V r V M k) )) r ). Thus we have N G V M k) )) V r r. Now we distinguish three cases. Case I: V M r) ) r +. Then we have G = V r + V M k) ) + rr + ) = f r) + k= which contradicts G f r). Case II: V M r) ) = r. Then we have V r r by Claim 3.9. If V M ) ) r + then we obtain G = V r + V M k) ) r + r + + r ) r = f r) + k= which contradicts G f r). Thus we may assume that V M k) ) = r for every k =... r. If V r r + then G = V r + V M k) ) r + + r r = f r) + k= so we can restrict ourselves to the following: V r = r and V M k) ) = r for every k =... r. For some k assume that there exists a vertex x k V M k) ) such that N G x k ) V M k) ) V M k) ) \ {x k }. Since d M k)x k ) = r and V M k) ) = r it holds that e G V r {x k }) = and then it must hold that e G V r V M k) )) r +. On the other hand e G V r V M k) )) r since V r = r a contradiction. Therefore V M k) ) G = Kr for every k =... r. Let V r = {z z... z r }. By the above arguments for every k =... r and l =... r there exist two distinct vertices x kl and y kl in V M k) ) such that x kl z l y kl z l e G V r V M k) )) F k). Now consider the vertices x kk and y kk for each k. Since V M k) ) G = Kr it is obvious that there exists a Hamilton path from x kk to y kk say P k and V M k) ) G P k is connected. Let C k be a cycle such that V C k ) = V P k ) {z k } and EC k ) = EP k ) {x kk z k y kk z k }. Moreover setting F = r k= Ck that is V F) = r k= V Ck ) = V G) and E F) = r k= ECk ) we can see that F is a -factor of G. As it is seen V M k) ) G P k is connected for every k =... r and N G V M k) )) N G V M l) )) = V r V r \ {z k z l } for k l and k l =... r. Moreover it holds that N G V M ) )) {z k z l } for mutually distinct k l. Thus G F is connected which implies that F is a NSF of G. This is a contradiction.

7 5544 Yu. Higuchi Y. Nomura / Discrete Mathematics ) Case III: V M r) ) = r. Then we have V r r by Claim 3.9. If k= V M k) ) 4r then we have G = V r + V M k) ) r + 4r + r )r ) = f r) + k= which contradicts G f r). So we may assume that k= V M k) ) 4r. Here we distinguish two subcases. Case III-i) Assume that V M ) ) = r and V M k) ) = r for each k = 3... r. If V r r then we have G = V r + V M k) ) r + r + r )r ) = f r) + k= which is a contradiction. So V r = r. On the other hand we have M k) = Kr since V M k) ) = r and M k) is r )-regular for k = 3... r; it holds that e G V r V M k) )) = r ). By using this together with the fact V r = r we have N G V M k) )) = V r and e G {v} V M k) )) = ) for k = 3... r and for any vertex v V r. Therefore V r V M k) ) F k) say C k) consists of some disoint cycles: for each k = 3... r where C k) C k) = m k = Ck) is a cycle with positive length in G. On the other hand Claim 3.9 tells us that N G V M ) )) V r r since V M ) ) = r. So choose and fix a vertex z N G V M ) )) V r ; here we may assume that z V C r) ). Then there exist two distinct vertices x r) and x r) in V M r) ) such that x r) z EC r) ) for = ; two such vertices can be chosen by ). Here remark x r) xr) EC r) ) and xr) xr) EG) since M r) = Kr. Define a new cycle C r) in G by ) V C r) ) = V Cr) ) \ {z } and E C r) ) = EC r) ) {xr) xr) } \ {x r) z z x r) }. Take two distinct vertices x r ) and x r ) in V M r ) ) such that x r ) z EG) for =. By the same argument as in Case II there exists a Hamilton path say P r ) from x r ) to x r ) since V M r ) ) G = Kr. It is obvious that V M r ) ) G P r ) is connected. Define a new cycle C r ) by V C r ) ) = V M r ) ) {z } and E C r ) ) = EP r ) ) {x r ) z z x r ) }. Moreover let F k) be a NSF of M k) for k =... r. Now we define a set of cycles F by F = r k= F k) C r ) C r) m r = C r) which is a -factor of G. Here V M k) ) G F is connected in G F for k =... r. In addition the following hold: N G F V M k) )) V r = V r for k = 3... r ; N G F V M r ) )) V r = V r \ {z }; N G F V M ) )) V r z ; N G F V M r) )) V r = {z }. Thus G F is connected that is G has a NSF a contradiction. Case III-ii) Assume that V M k) ) = r for each k =... r. If V r r + then we have G = V r + V M k) ) r + + rr ) = f r) + k= )

8 Yu. Higuchi Y. Nomura / Discrete Mathematics ) which is a contradiction. Moreover if V r = r then it follows from the same argument as in Case III-i) that G must have a NSF. This is also a contradiction. Then we may assume that V r = r. Again we distinguish two sub-subcases. Case III-ii)-a) Assume that N G V M i) )) V r for some i. Recall that M k) = Kr and e G V r V M k) )) = r ) for each k. Since V r = r there exists ust one vertex z V r such that N G z ) V M i) ) =. Furthermore we have r e G V r V M )) k) = rr ) 3) k= then there exists a vertex x 0 V r with x 0 z such that r e G {x 0 } V M )) k) = r and we have k= e G {x 0 } V M k) )) = 4) for every k =... r. On the other hand it holds that ) r e G {z } V M k) ) k= since d G z ) = r and V r \ {z } = r. Also it holds that since N G z ) V r 5) e G {z } r V M )) k) = e G {z } V M )) k) r ). k i k= Now we should remark there exists at most one vertex x V r such that e G {x} r k= V M k) )) ; if two distinct vertices satisfy the above then we have r e G V r V M )) k) + rr ) = rr ) r ) k= which contradicts 3) since r 3. In addition if such a vertex x exists then there exists ust one M k) such that e G {x} V M k) )) ; we replace z in this proof with such a vertex x. In other words we may say e G {z} k i V M k) )) for each z V r \ {z }. As is seen in ) the subgraph of F i) induced by V M i) ) V r \ {z }) consists of some cycles that is m i V M i) ) V r \ {z }) F i) = = C i) where C i) is a cycle for each. Letting C i) denote the cycle in F i) going through the vertex z we have V C i) ) V r = {z } and N i) C z ) V M s) ) for some s i). Suppose y V M s) ) and y V M t) ) such that s t and {y y } = N C i) we set N C i) z ). Then it must hold that V C i) ) \ {z }) V r by Claim 3.5 a contradiction. Here z ) = {y s) ys) }. It follows from Claim 3.9 and the fact V r = r that N G V M k) )) N G V M l) )) V r r 4 6)

9 5546 Yu. Higuchi Y. Nomura / Discrete Mathematics ) for every k l =... r. By 4) we have N G x 0 ) V M t) ) = for t i s); we set N G x 0 ) V M t) ) = {y t) yt) }. Since Ms) = Kr there exists a Hamilton path from y s) to y s) say Ps) ; it is obvious that M s) P s) is connected. Let us define a cycle C s) by V C s) ) = V M s) ) {z } and EC s) ) = EP s) ) {y s) z z y s) }. Similarly there exists a Hamilton path from y t) to y t) say Pt) ; it is also obvious that M t) P t) is connected. We define a cycle C t) by V C t) ) = V M t) ) {x 0 } and EC t) ) = EP t) ) {y t) x 0 x 0 y t) }. We may furthermore assume that x 0 V C i) ). Let N C i) x 0 ) = {y i) yi) }. Then we define a cycle C i) by ) V C i) ) = V Ci) ) \ {x 0} and E C i) ) = EC i) ) {yi) yi) } \ {y i) x 0 x 0 y i) }. Here remark that y i) yi) EG) and y i) yi) EC i) ). For every k i s t) let Ck) be a Hamilton cycle of M k) = Kr ; M k) C k) is connected. Now we define a set of cycles F by F = k ist C k) C s) C t) C i) m i =3 C i) which is a -factor of G. Remark the following: N G F V M s) )) V r = N G F V M t) )) V r r and N G F V M k) )) V r r for k i s t). Then for every k l i) N G F V M k) )) N G F V M l) )) V r r 4. On the other hand for every t) N G F x 0 ) V M ) ). Moreover N G z ) V r by 5); by 6) e G {z} k i V M k) )) for each z V r \ {z }. Thus it is shown that G F is connected a contradiction. Case III-ii)-b) Let us assume that N G V M k) )) = V r for each k =... r. Let pk) be pk) = {x V r N F k)x) V M k) ) = }. Then we have e G V r V M k) )) = r pk)) + pk) = 4r pk). On the other hand it holds that e G V r V M k) )) = r ) since M k) = Kr. Thus we have pk) = for each k; equivalently we have {x V r N F k)x) V M k) ) = } = r 4. 7) Let us recall that E V M k) ) G ) F k) = for each k =... r. It is easy to see that where C r) as m r V M r) ) V r F r) = Cr) F r) = = C r)... Cr) m r are mutually disoint cycles. The -factor F r) obtained in the Procedure can be expressed r k= m r H k) = C r) where H k) is a NSF of M k) = Kr for k =... r. It follows from 7) that the number of vertices v V r satisfying N F r )v) V M r ) ) = and N F r)v) V M r) ) = is greater than or equal to r 4.

10 Therefore there exists a vertex z V r such that Yu. Higuchi Y. Nomura / Discrete Mathematics ) N F r )z ) V M r ) ) = N F r)z ) V M r) ) =. Now we argue similarly as in Case III-i). Let N F r )z ) V M r ) ) = {x r ) x r ) } and N F r)z ) V M r) ) = {x r) xr) }. We choose a Hamilton path of Mr ) from x r ) to x r ) say P r ) ; then M r ) P r ) is connected. Define a cycle C r ) by V C r ) ) = V M r ) ) {z } and EC r ) ) = EP r ) ) {x r ) z z x r ) }. On the other hand we may assume that z V C r) ) since z V r m r x r) xr) EG) and x r) xr) F r). So define a cycle C r) by ) V C r) ) = V Cr) ) \ {z } and E C r) ) = EC r) ) {xr) xr) } \ {x r) z z x r) }. It is obvious that M r) {x r) F = r k= xr) H k) C r ) C r) m r } is connected. Now we define a set of cycles F by = C r) = V Cr) ). Here remark that which is a -factor of G. We can check the following: V M k) ) G F is connected for each k =... r; N G F V M k) )) = V r for each k =... r ; N G F V M r ) )) = V r \ {z }; N G F V M r) )) = {z }. This implies that G F is connected which is a contradiction. Consequently it is shown that Assumption.4 does not hold under Hypothesis.3. Using Lemma. together with this fact we complete the proof of Theorem. Acknowledgements We would like to thank Professors Katsuhiro Ota and Akira Saito for their valuable comments and suggestions. In addition we would like to thank also the anonymous referees who read our original manuscript carefully and gave us many useful comments. The first author was partially supported by JSPS under the Grant-in-Aid for Scientific Research No and The second author was partially supported by JSPS under the Grant-in-Aid for Scientific Research No and References [] R. Diestel Graph Theory second ed. Springer-Verlag 000. [] G.A. Dirac Some theorems on abstract graphs Proc. London Math. Soc. 95) [3] J. Petersen Die theorie der regulären graphs Acta. Math. 5 89) [4] Yu. Higuchi Y. Nomura Spectral structure of the Laplacian on a covering graph European J. Combin. in press). [5] B. Jackson Hamilton cycles in regular -connected graphs J. Combin. Theory Ser. B 9 980) [6] Y. Nomura private communication. [7] K. Ota private communication.