A 2factor in which each cycle has long length in clawfree graphs


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1 A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science (ITI) Charles University Univerzitní Plzeň Czech Republic. Department of Mathematical Information Science, Tokyo University of Science, 3 Kagurazaka, Shinjukuku, Tokyo 6860, Japan. 3 Department of Mathematics, College of Science and Technology, Nihon University, Tokyo , Japan. Abstract A graph G is said to be clawfree if G has no induced subgraph isomorphic to K,3. In this article, we prove that every clawfree graph G with minimum degree δ(g) 4 has a factor in which each cycle contains at least δ(g) vertices and every connected clawfree graph G with δ(g) 3 has a factor in which each cycle contains at least δ(g) vertices. For the case where G is connected, the lower bound on the length of a cycle is best possible. Keywords: factor, Clawfree graph, Minimum degree AMS Subject Classification: 05C70 Introduction In this paper, we consider finite graphs. A simple graph means an undirected graph without loops or multiple edges. A multigraph may contain multiple edges but no loops. Let G be a graph. For a vertex v of G, the degree of v in G is the number of edges incident with v. Let V (G), E(G) and δ(g) be the vertex set, the edge set and the minimum degree of G, respectively. We refer to the number of vertices of G as the order of G and denote it by G. A graph G is said to be clawfree if G has no induced subgraph isomorphic to K,3 (here K,3 denotes the An extended abstract published in Eurocomb0, Electronic Notes in Discrete Mathematics Vol. 38 (0) address: This work was supported by No. M of the Czech Ministry of Education address: This work was supported by GrantinAid for Young Scientists (B) ( ) address: This work was supported by KAKENHI (5405)
2 complete bipartite graph with partite sets of cardinalities and 3, respectively). A factor of G is a spanning subgraph of G in which every component is a cycle. It is a wellknown conjecture that every 4connected clawfree graph is Hamiltonian due to Matthews and Sumner []. Since a Hamilton cycle is a connected factor, there are many results on factors of clawfree graphs. For instance, results of both Choudum and Paulraj [4] and Egawa and Ota [5] imply that a moderate minimum degree condition already guarantees that a clawfree graph has a factor. Theorem A ([4, 5]) Every clawfree graph G with δ(g) 4 has a factor. For connected clawfree graphs, the third author [5] improved the condition on minimum degree as follows. Theorem B ([5]) Every connected clawfree graph G with δ(g) 3 has a factor. In particular, in [], it is shown that Matthews and Sumner s conjecture and the statement that every 4connected clawfree graph of order n has a factor with at most f(n) components (here f(n) is a function of n with the property that lim n f(n)/n = 0) are equivalent. This indicates that it is important to study about the number of components of factors in clawfree graphs to solve Matthews and Sumner s conjecture. Concerning the upper bound on the number of components, Broersma, Paulusma and the third author [3] and Jackson and the third author [8] proved the following, respectively (other related results can be found in [6, 9, 0,, 5]). Theorem C ([3]) Every clawfree graph G with δ(g) 4 has a factor with at most max { G 3 δ(g), } components. Theorem D ([8]) Every connected clawfree graph G with δ(g) 4 has a factor with at most G + 4 components. In this paper, we approach this problem in terms of the lengths of cycles in a factor of a clawfree graph. More precisely, we consider the functions f : {d Z d 4} {m Z m 3} and f : {d Z d 3} {m Z m 3} defined as follows: for an integer d with d 4, let f (d) := max{m every connected clawfree graph with minimum degree d has a factor in which each cycle contains at least m vertices}, and for an integer d 3, let f (d) := max{m every connected clawfree graph with minimum degree d has a factor in which each cycle contains at least m vertices}. Concerning the determinations of f (d) and f (d), we will show the following two results in Sections 3 and 4. Theorem Every clawfree graph G with δ(g) 4 has a factor in which each cycle contains at least δ(g) vertices.
3 G u H K d+ K d+ K d+ v u H K d+ K d+ K d+ v u 3 H 3 K d+ K d+ K d+ v 3 Figure : The graph G Theorem Every connected clawfree graph G with δ(g) 3 has a factor in which each cycle contains at least δ(g) vertices. As a corollary of Theorem, we can get the following which is slightly stronger than Theorems C and D for connected clawfree graphs. Corollary 3 Every connected clawfree graph G with δ(g) 3 has a factor with at most G δ(g) components. Moreover, in Theorem, the lower bound on the length of a cycle is best possible because we can construct a connected clawfree graph G such that for each factor of G, the minimum length of cycles in it is at most δ(g) as follows: let d 3 be an integer, and let H, H and H 3 be graphs with the connectivity one such that each block of H i ( i 3) is a complete graph of order d +. For each i 3, let u i and v i be vertices contained in distinct end blocks of H i, respectively, and u i and v i are not cut vertices of H i. Let G be a graph obtained from H H H 3 by joining u i and u i+ for i 3 and joining v i and v i+ for i 3, where let u 4 := u and v 4 := v, see Figure (here K m denotes the complete graph of order m). Then G is a connected clawfree graph with δ(g ) = d, and for each factor of G, the minimum length of cycles in it is at most d. So this together with Theorem implies that f (d) = d holds. On the other hand, we can obtain slightly stronger statement than Theorem if we consider only line graphs of trees as follows, which will be shown in Section 5 (here a cliquefactor is a spanning subgraph in which every component is a clique). 3
4 Theorem 4 The line graph G of a tree has a cliquefactor in which each clique contains at least vertices. δ(g)+ In view of Theorem 4, the following statement might hold. Conjecture 5 Every clawfree graph G with δ(g) 4 has a factor in which each cycle contains at least δ(g)+ vertices. In Conjecture 5, the lower bound on the length of a cycle is best possible if it is true. So this together with Theorem implies that d f (d) d+ holds. In section 5, we give a graph showing the sharpness of the lower bound in Conjecture 5. Now we have that f (d) is completely determined and f (d) is almost determined, in particular, f (d) = d and d f (d) d+ hold even if we consider only graphs with sufficiently large orders compared to d (see Figure and see the second paragraph and Figure 4 in Section 5). Therefore it would be natural to consider the case where the connectivity is at least 3 as the next step. If Matthews and Sumner s conjecture is true, we can always take Hamilton cycles in 4connected clawfree graphs. So what will happen when the connectivity is just 3? For instance, concerning the number of components of factors in 3connected clawfree graphs, Kužel, Ozeki and the third author [0] proved that every 3connected clawfree graph G has a factor with at most max { G } δ(g)+, components. Furthermore, most recently, Ozeki, Ryjáček and the third author [] proved that every 3connected clawfree graph G has a factor with at most 4 G 5(δ(G)+) + 5 components. In view of this result, one might expect that the coefficient of d in the lower bound on the length of a cycle would be greater than for 3connected clawfree graphs. Problem Determine f 3 (d) := max{m every sufficiently large 3connected clawfree graph with minimum degree d has a factor in which each cycle contains at least m vertices}. In particular, is there an constant c > such that f 3 (d) cd holds? It is of course important to consider about the case of 4connected clawfree graphs. As mentioned above, if there exists a function f(n) of n such that lim n f(n)/n = 0 and every 4connected clawfree graph of order n has a factor with at most f(n) components, then every 4connected clawfree graph is Hamiltonian, i.e., to solve Matthews and Sumner s conjecture, it suffices to show the existence of a factor with small number of components (not necessarily component). Therefore it could be also another possible approach to study about the lengths of cycles in factors of 4connected clawfree graphs. In addition, for a result similar to Theorems, and 4 concerning pathfactors (here a pathfactor is a spanning subgraph in which every component is a path), Ando, Egawa, Kaneko, Kawarabayashi and Matsuda [] proved that every clawfree graph G has a pathfactor in which 4
5 each path contains at least δ(g) + vertices, and they conjectured that every connected clawfree graph G has a pathfactor in which each path contains at least 3δ(G) + 3 vertices. In the rest of this section, we prepare notation which we use in subsequent sections. Let G be a graph. We denote the number of edges of G by e(g) and the line graph of G by L(G). For X V (G), we let G[X] denote the subgraph induced by X in G, and let G \ X := G[V (G) \ X]. If H is a subgraph of G, then let G \ H := G \ V (H). A subset X of V (G) is called a independent set of G if G[X] is edgeless. Let H and H be subgraphs of G or subsets of V (G), respectively. If H and H have no any common vertex in G, we define E G (H, H ) to be the set of edges of G between H and H, and let e G (H, H ) := E G (H, H ). For a vertex v of G, we denote by N G (v) and d G (v) the neighborhood and the degree of v in G, respectively. Let o(g) the set of vertices with odd degree in G. For a positive integer l, we define V l (G) := {v V (G) d G (v) = l}, and let V l (G) := m l V m(g) and V l (G) := m l V m(g). Let e = uv E(G). We denote by V (e) the set of end vertices of e, i.e., V (e) = {u, v}. The edge degree of e in G is defined by the number of edges incident with e, and is denoted by ξ G (e), i.e., ξ G (e) = {f E(G) f e, V (f) V (e) }. Note that if G is a simple graph, then ξ G (e) = e G (V (e), G \ e) = d G (u) + d G (v). Let ξ(g) be the minimum edge degree of G. Outline of the proofs of Theorems, and 4 In this section, we give an outline of the proofs of Theorems, and 4. To prove Theorems and, we use Ryjáček closure which is defined as follows. Let G be a clawfree graph. For each vertex v of G, G[N G (v)] has at most two components; otherwise G contains a K,3 as an induced subgraph. If G[N G (v)] has two components, both of them must be cliques. In the case that G[N G (v)] is connected, we add edges joining all pairs of nonadjacent vertices in N G (v). The closure cl(g) of G is a graph obtained by recursively repeating this operation, as long as this is possible. In [3], it is shown that cl(g) is uniquely defined and cl(g) is the line graph of some trianglefree simple graph. On the other hand, in [4, Theoerm 4], Ryjáček, Saito and Schelp proved that for any vertexdisjoint cycles D,..., D q in cl(g), G has vertexdisjoint cycles C,..., C p with p q such that q i= V (D i) p i= V (C i). By modifying the proof, we can easily improve this result as follows (see [0, Lemma 3]): for any vertexdisjoint cycles D,..., D q in cl(g), G has vertexdisjoint cycles C,..., C p with p q such that for each j with j q, there exists i with i p such that V (D j ) V (C i ). These imply that for a clawfree graph G, G has a factor in which each cycle contains at least m vertices if and only if cl(g) has a factor in which each cycle contains at least m vertices. By this fact, to prove Theorems and, it suffices to show that the following hold (here a graph H is called essentially kedgeconnected if e(h) k + and H \ F has at most one component which contains an edge 5
6 for every F E(H) with F < k. It is easy to see that for a graph H such that L(H) is not complete, H is essentially kedgeconnected if and only if L(H) is kconnected). Theorem 6 Let G be the line graph of some trianglefree simple graph and δ(g) 4. Then G has a factor in which each cycle contains at least δ(g) vertices. Theorem 7 Let G be the line graph of some essentially edgeconnected trianglefree simple graph and δ(g) 3. Then G has a factor in which each cycle contains at least δ(g) vertices. Moreover, to prove Theorems 7 and 4, we consider about some particular set of subgraphs. Now let H be a multigraph. We call a subgraph S of H a star if S consists of a vertex (called a center) and edges incident with the center. So a star in this paper is not necessary a tree. A closed trail T in H is called a dominating closed trail if H \ T is edgeless. It is known that for a connected multigraph H with e(h) 3, H has a dominating closed trail if and only if L(H) is Hamiltonian (see [7]). A set D is called a cover set of E(H) if (i) D is a set of edgedisjoint connected subgraphs in H such that D D E(D) = E(H), and (ii) for each D D, D is a star or D has a dominating closed trail. Let D be a cover set of E(H). If D is a star for all D D, then D is called a star cover set of E(H). Note that for each D D, if D is a star, then L(D) is a clique. Therefore, to prove Theorems 7 and 4, it suffices to show that the following hold, respectively. Theorem 8 Let G be an essentially edgeconnected trianglefree simple graph with ξ(g) 3. Then G has a cover set D of E(G) such that e(d) ξ(g) for all D D. Theorem 9 A tree G with ξ(g) has a star cover set S of E(G) such that e(s) ξ(g)+ for all S S. The following lemma is used to find dominating closed trails in the proof of Theorem 8. Lemma ([5]) If G is an essentially edgeconnected graph with ξ(g) 3, then there exists a set T of mutually vertexdisjoint closed trails in G such that V 3 (G \ V (G)) T T V (T ). 3 Proof of Theorem 8 If G is a star, then the assertion clearly holds. Thus we may assume that G is not a star. Then since G is essentially edgeconnected, we have δ(g \ V (G)). By Lemma, there exists a set T of mutually vertexdisjoint closed trails in G such that V 3 (G \ V (G)) T T V (T ). 6
7 G T G v T T v T T v T Figure : The graph G Since δ(g \ V (G)) and V 3 (G \ V (G)) T T V (T ), v V (G \ V (G)) for all v V 3 (G) \ T T V (T ). This implies that for each v V 3(G) \ T T V (T ), there exist exactly two vertices u, u N G (v) with u u such that u i V (G) for i =, and N G (v) \ {u, u } V (G); (3.) in particular, N G (v) \ {u, u } = since v V 3 (G), that is, N G (v) V (G), and hence N G (v) \ {u, u } d G (v) ξ(g). Let G be a graph obtained from G by contracting an induced subgraph G[V (T )] of G to a vertex v T for each T T (note that G may be a multigraph, see Figure ). Let X := {v T T T }. Note that X = T since T is a set of mutually vertexdisjoint closed trails in G. By the definition of G, d G (v) = d G (v) for all v V (G ) \ X. (3.) By (3.) and since V (G) ( T T V (T )) =, V (G) = V (G ) \ X. (3.3) By the definition of G and (3.) and since ξ(g) 3, we also have that V (G ) \ X is an independent set of G. To show the existence of a cover set D of E(G), we find a mapping ϕ : E(G ) V (G ) so that (M) ϕ(e) = u or ϕ(e) = v for all e = uv E(G ), (M) ϕ (v) = 0 for all v V (G ) \ X, (M3) ϕ (v) ξ(g) for all v V 3 (G ) \ X, (M4) ϕ (v T ) + e(g[v (T )]) ξ(g) for all v T X. 7
8 Suppose that there exists such a mapping ϕ. Let S := {S v v V 3 (G ) X} be a set of stars such that for each S v S, S v is a star consisting of a vertex v (as the center) and the edges in ϕ (v). Then by the conditions (M) and (M), S is a star cover set of E(G ). Furthermore by the conditions (M3) and (M4), e(s v ) = ϕ (v) ξ(g) for each v V 3 (G ) \ X and e(s vt ) = ϕ (v T ) ξ(g) e(g[v (T )]) for each v T X. Hence by the definition of G, and by considering a subset of E(G) corresponding to E(S v ) for each v V 3 (G ) X and replacing v T with G[V (T )] for each v T X, we can find a cover set D of E(G) such that e(d) ξ(g) for all D D. Thus it suffices to show the existence of such a mapping ϕ. Let G := G \ (V (G ) \ X). To show the existence of a mapping ϕ satisfying the conditions (M) (M4), we define a mapping ϕ : E(G ) V (G ) as follows. Since the number of vertices of odd degree is even in G, there exists a collection of paths P,..., P l in G such that each vertex in o(g ) appears in the set of end vertices of them exactly one (note that l = o(g ) /). By considering the symmetric difference of them, we may assume that P,..., P l are pairwise edgedisjoint. For each i l, let x i xi xi 3... xi P i xi P i := P i, and let e i j := xi j xi j+ for each j P i, and we define ϕ (e i j ) := xi j+ for each j P i. Let G := G \ l i= E(P i). By the definitions of P,..., P l, we have o(g ) =. Hence the edges of G can be covered by pairwise edgedisjoint cycles. For each cycle, written by y y y 3... y m y m (= y ), we define ϕ (e j ) := y j+, where e j := y j y j+ for each j m. Then by the definition of ϕ, for each v V (G ), ( (dg ϕ (v) (d G (v) )/ = (v) (N G (v) V (G )) \ X ) ) /. (3.4) Now we define a mapping ϕ : E(G ) V (G ) as follows: for each e = uv E(G ), we let { u if v V (G ) \ X ϕ(e) := ϕ (e) otherwise. Since V (G ) \ X is an independent set of G, ϕ is well defined. By the definitions of ϕ and ϕ, we can easily see that ϕ satisfies the conditions (M) and (M). So we show that ϕ satisfies the conditions (M3) and (M4). Since (V (G ) \ X) V (G ) = by the definition of G, it follows from the definition of ϕ that ϕ (v) ϕ (v) + (N G (v) V (G )) \ X for all v X. (3.5) We first show that ϕ satisfies the condition (M3). Let v V 3 (G ) \ X. Then by (3.) and the definitions of G and X, v V 3 (G) \ T T V (T ). Hence by (3.), there exist exactly two vertices u, u N G (v) with u u such that u i V (G) for i =,, N G (v)\{u, u } V (G) and N G (v) \ {u, u } ξ(g). Therefore by (3.), (3.3) and the definition of G, there exist two distinct edges vu, vu E(G ) such that u i V (G ) X for i =,, N G (v) \ {u, u } 8
9 V (G ) \ X and N G (v) \ {u, u } ξ(g) (possibly u = u, i.e., vu and vu may be parallel edges in G ). If u i V (G ) \ X for some i = or, then by the definition of ϕ, we have ϕ (v) (N G (v) V (G )) \ X + (N G (v) V (G )) \ X N G (v) \ {u, u } + (ξ(g) )+ = ξ(g). Thus we may assume that u i V 3(G ) X for i =,. Note that by the definition of G, u i V (G ) for i =,. Since N G (v)\{u, u } V (G )\X, we have v V (G ) and vu i E(G ) for i =,. Hence by the definition of ϕ, ϕ (vu i ) = v and ϕ (vu 3 i ) = u 3 i for some i = or. Then by the definition of ϕ and since {v, u, u } V 3(G ) X, ϕ(vu i ) = v for some i = or. Therefore ϕ (v) (N G (v) V (G ))\X + {vu i } N G (v)\{u, u } + (ξ(g) ) + = ξ(g) holds for some i = or. Thus ϕ satisfies the condition (M3). We next show that ϕ satisfies the condition (M4). Let T T. Since G is trianglefree, all cycle which is contained in T has order at least four. Hence there exist two independent edges e and e in T. For each i =,, let ɛ i := {e E(G[V (T )]) e ei, V (e) V (e i ) }. Since G is trianglefree, {e E(G[V (T )]) V (e) V (ei ) for i =, }. Hence by the definition of ɛ i, e(g[v (T )]) (ɛ + ɛ ) + {e, e } = ɛ + ɛ. (3.6) We also have e G (V (e i ), G \ T ) ξ(g) ɛ i for each i =,. Hence since d G (v T ) = e G (T, G \ T ) by the definition of G and since E G (V (e ), G \ T ) E G (V (e ), G \ T ) =, we obtain d G (v T ) = e G (T, G \ T ) e G (V (e ), G \ T ) + e G (V (e ), G \ T ) ξ(g) (ɛ + ɛ ). (3.7) Then by (3.4), (3.5), (3.6) and (3.7), ϕ (v T ) ϕ (v T ) + (N G (v T ) V (G )) \ X ( d G (v T ) (N G (v T ) V (G )) \ X ) / + (N G (v T ) V (G )) \ X ( ξ(g) (ɛ + ɛ ) ) / = ξ(g) ( (ɛ + ɛ ) + ) / ξ(g) ( e(g[v (T )]) + ) /. Since e(g[v (T )]) 4, we get ϕ (v T ) +e(g[v (T )]) ξ(g) ( e(g[v (T )])+ ) /+e(g[v (T )]) ξ(g) + ( e(g[v (T )]) ) / > ξ(g). Thus ϕ satisfies the condition (M4). This completes the proof of Theorem 8. 9
10 T B 0 = {B 0 } C 0 B C B C C m B m Figure 3: Subsets B 0,..., B m and subsets C 0,..., C m 4 Proof of Theorem 6 To prove Theorem 6, we use Theorem and the following proposition. Proposition 0 Let G be a line graph with δ(g) 4. Then there exists a set H of mutually vertexdisjoint subgraphs in G such that H H V (H) = V (G), and H is a connected clawfree graph and δ(h) δ(g) for all H H. Proof of Proposition 0. It suffices to consider the case where G is connected. By the assumption, we may assume that the connectivity of G is one. Let B be the set of blocks of G and C be the set of cut vertices of G. We consider a block cut tree T of G, i.e., V (T ) = B C and E(T ) = {Bv B B, v V (B) C}. Since G is a line graph, for each v V (G), G[N G (v)] has a cliquefactor with at most components. (4.) Hence d T (v) = for all v C. Let B 0 B be a root of T. We consider T as an oriented tree from B 0 to leaves, and denote it T. For each v C, we let B v + (resp. B v ) denote a successor (resp. a predecessor) of v along T (note that B v + and B v are uniquely defined because d T (v) =, and note also that B v +, B v B). For each B B \ {B 0 }, we let v B denote a predecessor of B along T (note that v B is uniquely defined). We define subsets B 0,..., B m of B and subsets C 0,..., C m of C inductively by the following procedure. First let B 0 := {B 0 }, and let C 0 := {v C B v = B 0 }. Now let i, and assume that we have defined B 0,..., B i and C 0,..., C i. If C i, then let B i := {B v + v C i }, and let C i := {v C B v B i }; if C i =, we let m := i and terminate the procedure (see Figure 3). Then it follows from 0
11 the definitions of B 0,..., B m and C 0,..., C m that B is disjoint union of B 0,..., B m and C is disjoint union of C 0,..., C m. We define a mapping ϕ : C B inductively as follows. First for each v C 0, let { B ϕ(v) := v if N G (v) V (B v ) δ(g). B v + otherwise Next for each v C i with i m, let ϕ(v) := B v if ϕ(v ) = B Bv v and N G(v) V (B v ) δ(g) B v if ϕ(v ) B Bv v and N G(v) V (B v ) δ(g)+ B v B v + if ϕ(v ) B Bv v, N G(v) V (B v ) = δ(g) and vv Bv otherwise / E(G). Let H := {G[(V (B) \ C) ϕ (B)] B B}. We show that H is a desired set. By the definition of H, it is easy to check that H is a set of mutually vertexdisjoint subgraphs in G such that H H V (H) = V (G) and H is clawfree for all H H. Let H H, and let H = G[(V (B) \ C) ϕ (B)] for some B B i with 0 i m. We first show that δ(h) δ(g) Case. B B 0, i.e., B = B 0.. Let u V (H). Assume for the moment that u ϕ (B 0 ). Then by the definition of ϕ, N G (u) V (B 0 ) δ(g). Since G[N G(u) V (B 0 )] is a complete graph by (4.), N G (v) V (B 0 ) δ(g) for all v N G (u) V (B 0 ). Hence by the definition of ϕ, ϕ(v) = B 0 for all v N G (u) V (B 0 ) C. Since H = G[(V (B 0 ) \ C) ϕ (B 0 )], this implies that (N G (u) V (B 0 )) {u} V (H), and hence d H (u) = N G (u) V (B 0 ) δ(g). Thus we may assume that u V (B 0) \ C. Then d B0 (u) = d G (u) δ(g). Hence by (4.), there exists a complete subgraph F of G[N B0 (u) {u}] such that u V (F ) and F δ(g)+. Then N G (v) V (B 0 ) F δ(g) for all v V (F ). Hence by the definition of ϕ, ϕ(v) = B 0 for all v V (F ) C. Since H = G[(V (B 0 )\C) ϕ (B 0 )], this implies that V (F ) V (H), and hence d H (u) F δ(g). Case. B B i with i m and u ϕ (B). Assume for the moment that u = v B, or u v B and ϕ(v B ) = B. Then by the definition of ϕ, N G (u) V (B) δ(g). Since G[N G(u) V (B)] is a complete graph by (4.), N G (v) V (B) δ(g) for all v N G (u) V (B). Since ϕ(v B ) = B, it follows from the definition of ϕ that ϕ(v) = B for all v N G (u) V (B) C. Since H = G[(V (B) \ C) ϕ (B)], this implies that (N G (u) V (B)) {u} V (H), and hence d H (u) = N G (u) V (B) δ(g). Thus we may
12 assume that u v B and ϕ(v B ) B. Then by the definition of ϕ, (i) N G (u) V (B) δ(g)+, or (ii) N G (u) V (B) = δ(g) and uv B / E(G). Since G[N G (u) V (B)] is a complete graph by (4.), for each v N G (u) V (B), N G (v) V (B) δ(g)+, or N G (v) V (B) = δ(g) and vv B / E(G). Hence by the definition of ϕ, ϕ(v) = B for all v N G (u) V (B) (C \ {v B }). Since H = G[(V (B) \ C) ϕ (B)], this implies that ( (N G (u) V (B)) {u} ) \ {v B } V (H), and hence d H (u) (N G (u) V (B)) \ {v B } δ(g). Case 3. B B i with i m and u / ϕ (B). Then u V (B) \ C, and hence d B (u) = d G (u) δ(g). Therefore by (4.), there exists a complete subgraph F of B[N B (u) {u}] such that (i) u V (F ) and F δ(g)+3, or (ii) u V (F ), F = δ(g)+ and v B / V (F ). Then since F is a complete graph, for each v V (F ), N G (v) V (B) δ(g)+, or N G (v) V (B) = δ(g) and vv B / E(G). Hence by the definition of ϕ, ϕ(v) = B for all v V (F ) (C \ {v B }). Since H = G[(V (B) \ C) ϕ (B)], this implies that V (F ) \ {v B } V (H), and hence d H (u) V (F ) \ {u, v B } δ(g) By Cases, and 3, d H (u) δ(g) We next show that H is connected. Suppose that H is not connected. Since δ(h) δ(g).. Since u is an arbitrary vertex in H, δ(h) δ(g), H 3. Hence there exist two distinct subgraphs F and F of H such that V (F j ) \ V (F 3 j ) for j =,, and V (F ) V (F ), V (H) = V (F ) V (F ) and E G (F \ F, F \ F ) =. Since B H 3, we also have that B is connected. Hence V (B \ H) = (V (B) C) \ ϕ (B). Let w V (F ) V (F ) if V (F ) V (F ) ; otherwise, let w be an arbitrary vertex in B \ H. Let U := {u V (B \ H) N G (u) (V (F ) \ {w}) }. Since B is connected, U. If there exists u U such that N G (u) (V (F ) \ {w}), then G[V (F F B u +) {u}] contains K,3 as an induced subgraph because E G (B \ {u}, B u +) = since u is a cut vertex of G, a contradiction. Thus N G (u) (V (F )\{w}) = for all u U. Hence since B is connected, there exist u U and v V (B)\(V (H) U) such that uv E(G). Then G[V (F B u +) {u, v}] contains K,3 as an induced subgraph because E G (B \ {u}, B u +) = since u is a cut vertex of G, a contradiction again. Thus H is connected. This completes the proof of Proposition 0. Proof of Theorem 6. Let G be as in Theorem 6. By Theorem A, we may assume that δ(g) 6. By Proposition 0, there exists a set H of mutually vertexdisjoint subgraphs in G such that H H V (H) = V (G), and H is a connected clawfree graph and δ(h) δ(g) 3 for all H H. Then by Theorem, H has a factor in which each cycle contains at least δ(h) δ(g) vertices for all H H. Since V (G) = H H V (H) (disjoint union), this implies that G has a factor in which each cycle contains at least δ(g) vertices..
13 G K d K (d+)/ K (d+)/ K d K d + + Y + + K (d+)/ K (d+)/ y + + x X Y X K d K (d+)/ K (d+)/ K d K d K d K d Figure 4: The graph G 5 Proof of Theorem 9 As is mentioned in Section, in this section, we give a graph showing the sharpness of the lower bound on the length of a cycle in Conjecture 5, and we prove Theorem 9. The graph G illustrated in Figure 4 (here, in Figure 4, + means the join of vertices) is a clawfree graph with minimum degree d ( 4), and for each factor of G, the minimum length of cycles in it is at most d+, and G has a factor such that the minimum length of cycles in it is just d+. To see this, let C be a set of mutually vertexdisjoint cycles in G such that C C C is a factor of G. If x X and y Y are contained in same cycle in C, then by the definition of G, the maximum size of the cycle containing both x and y is d+. Hence by the symmetry, we may assume that for all x X X and all y Y Y, the cycle containing x and the cycle containing y in C are distinct. Then by the definition of G, there exists a subset C of C such that C C V (C) = X or C C V (C) = X. Since X = X = d+, the minimum length of cycles in C is at most d+. Moreover, by these arguments, we can take a set C so that the minimum length of cycles in C is just d+. (Note that by adding X, Y, X and Y iteratively, we can construct a sufficiently large order graph.) To prove Theorem 9, we prove slightly stronger statement than Theorem 9 as follows (note that for each tree G with ξ(g), there exist vertices u V ξ(g) + (G) and u N G (u ) such that N G (u ) \ {u } V (G), and hence Theorem implies Theorem 9). Theorem Let G be a tree with ξ(g), and let u V l+ (G) and u N G (u ) such 3
14 T 0 G G i T 0 X u T 0 u u x x x m x i T T T m T i Figure 5: The tree G i that N G (u ) \ {u } V (G), where let l := ξ(g). Then the following hold. (I) If u V l (G), then G has a star cover set S of E(G) such that e(s) ξ(g)+ for all S S, and there exists S u S such that u u E(S u ) and u is the center of S u. (II) If u V l+ (G), then G has two star cover sets S and S of E(G) such that for each i =,, the following hold: (i) e(s) ξ(g)+ for all S Si. (ii) there exists S i S i such that u u E(S i ) and u i is the center of S i. Proof of Theorem. We prove Theorem by induction on e(g). If N G (u ) = V (G), then S := {G} is a star cover set of E(G) such that S satisfies the statement (I) in Theorem. Thus we may assume that N G (u ) V (G). Write N G (u ) \ {u } = {x, x,..., x m }. Let X be a subset of N G (u ) \ {u } such that N G (u ) \ ({u } X) = ξ(g) (note that d G (u ) ξ(g) + since d G (u ) l + and N G (u ) \ {u } V (G)). Let T 0, T,..., T m be components of G \ {u } such that T 0 contains a vertex u, and T i contains a vertex x i for each i m, and let T 0 := T 0 \ X. We construct m trees G, G,..., G m as follows. For each i m, let V (G i ) := V (T 0 ) V (T i ) and E(G i ) := E(T 0 ) E(T i ) {u x i } (see Figure 5). Then by the definitions of T 0, T,..., T m and G,..., G m, for each i m, E(G i ) < E(G) and d Gi (v) = d G (v) for all v V (G i ) \ {u }. (5.) 4
15 Since e(t0 ) = ξ(g), we also have ξ(g i ) = ξ(g) for each i m. (5.) We divide the proof into three cases according as u V l (G), or u V l+ (G) and N G (u ) V l+ (G), or u V l+ (G) and N G (u ) V l+ (G). Case : u V l (G). Then N G (u ) V l+ (G) (otherwise, ξ G (u x) < ξ(g) for x N G (u ) V l (G), a contradiction). Fix i with i m. Then by (5.) and (5.) and since e(t 0 ) = ξ(g), {u, x i } V l+ (G i ) = V ξ(gi ) + (G i). Therefore by the induction hypothesis and (5.), G i has a star cover set S i of E(G i) such that ξ(gi ) + ξ(g) + e(s) = for all S S i, (5.3) and there exists S xi S i such that u x i E(S xi ) and x i is the center of S xi. Note that T 0 S i because S x i S i and e(t 0 ) = ξ(g i) by (5.). Since i is an arbitrary integer with i m, there exists such a star cover set S i of E(G i) for all i with i m. For each i m, let Sx i be a star consisting of a vertex x i (as the center) and the edges in (E(S xi ) \ {u x i }) {u x i }, and let Su be a star consisting of a vertex u (as the center) and the edges in E(T 0 ) {u u }. Then by the definitions of Sx i and Su, e(sx i ) = e(s xi ) for each i m and e(su ) e(t0 ) = ξ(g). Hence by (5.3) and the definitions of S x i and Su, S := m ( i= S i \ {S xi, T0 }) {Su, Sx, Sx,..., Sx m } is a desired star cover set of E(G) (note that Su is a star in S such that u u E(Su ) and u is the center of Su ). Thus the statement (I) in Theorem holds. Case : u V l+ (G) and N G (u ) V l+ (G). Then we may assume that there exists an integer s with 0 s m such that {x,..., x s } V l+ (G) and {x s+,..., x m } V l (G). Since N G (u ) V l (G), we have ξ(g) + m (otherwise, ξ G (u x i ) < ξ(g) for s + i m, a contradiction). Fix i with i m. Then by (5.) and (5.), x i V l+ (G i ) = V ξ(gi ) + (G i) if i s; otherwise, x i V l (G i ) = V ξ(gi ) (Gi ). Therefore by the induction hypothesis, G i has a star cover set S i of E(G i) such that ξ(gi ) + ξ(g) + e(s) = for all S S i, (5.5) and there exists S i u S i such that u x i E(S i u ) and u is the center of S i u. Since i is an arbitrary integer with i m, there exists such a star cover set S i of E(G i) for all i 5 (5.4)
16 with i m. For each i m, let Su i := {S S i u is the center of S}. Then S Su i E(S) = E(T0 ) {u x i } for each i m because Su i S i. Let Su be a star consisting of a vertex u (as the center) and the edges in E(T 0 ) {u u }, Su be a star consisting of a vertex u (as the center) and the edges in E(T 0 ), Su be a star consisting of a vertex u (as the center) and the edges in {u x i i m}, and S u be a star consisting of a vertex u (as the center) and the edges in {u x i i m} {u u }. Then by the definitions of S u, S u, S u and Su, e(su ) > e(su ) e(t0 ) = ξ(g) and e(s u ) = e(su ) + = m +. Hence by (5.4), (5.5) and the definitions of Su, Su, Su E(T0 ) {u x i } for each i m, we have that S := m S := m i= and S u and since S S i u E(S) = i= ( S i \ S i u ) {S u, S u } and ( S i \ S i u ) {S u, S u } are desired star cover sets of E(G) (note that S u is a star in S such that u u E(Su ) and u is the center of Su, and Su is a star in S such that u u E(Su ) and u is the center of Su ). Thus the statement (II) in Theorem holds. Case 3: u V l+ (G) and N G (u ) V l+ (G). Fix i with i m. Then by (5.), (5.) and the assumption of Case 3, {u, x i } V l+ (G i ) = V ξ(gi ) + (G i). Therefore by the induction hypothesis and (5.), G i has two star cover sets S i and S i of E(G i) such that ξ(gi ) + ξ(g) + e(s) = for all S S i S i, (5.6) and there exists S i u S i such that u x i E(S i u ) and u is the center of S i u, and there exists S xi S i such that u x i E(S xi ) and x i is the center of S xi. Since i is an arbitrary integer with i m, there exist such star cover sets S i and S i of E(G i) for all i with i m. We first find a star cover set S of E(G). Note that T0 S i for each i m because e(t0 ) = ξ(g) and S x i S i. For each i m, let S x i be a star consisting of a vertex x i (as the center) and the edges in (E(S xi ) \ {u x i }) {u x i }, and let Su be a star consisting of a vertex u (as the center) and the edges in E(T 0 ) {u u }. Then by the definitions of Sx i and Su, e(sx i ) = e(s xi ) for each i m and e(su ) e(t0 ) = ξ(g). Hence by (5.6) and the definitions of Sx i and Su, S := m ( i= S i \ {S xi, T0 }) {Su, Sx, Sx,..., Sx m } is a desired star cover set of E(G) (note that Su is a star in S such that u u E(Su ) and u is the center of S u ). We next find a star cover set S of E(G). For each i m, let S i u := {S S i u is the center of S}. Then S S i u E(S) = E(T 0 ) {u x i } for each i m because S i u S i. Let Su be a star consisting of a vertex u (as the center) and the edges in E(T 0 ) and Su be a star consisting of a vertex u (as the center) and the edges in {u x i i m} {u u }. Then by the definitions of Su and Su and the assumption of Case 3, e(su ) e(t0 ) = ξ(g) and e(su ) = m + = d G (u ) l + = ξ(g) +. Hence by (5.6) and the definitions of 6
17 Su and Su and since S Su i E(S) = E(T0 ) {u x i } for each i m, we have that S := m ( i= S i \ Su i ) {S u, Su } is a desired star cover set of E(G) (note that Su is a star in S such that u u E(Su ) and u is the center of Su ). Thus the statement (II) in Theorem holds. References [] K. Ando, Y. Egawa, A. Kaneko, K. Kawarabayashi and H. Matsuda, Path factors in clawfree graphs, Discrete Math. 43 (00) [] H. J. Broersma, M. Kriesell and Z. Ryjáček, On factors of 4connected clawfree graphs, J. Graph Theory 37 (00), [3] H. J. Broersma, D. Paulusma and K. Yoshimoto, Sharp upper bounds for the minimum number of components of factors in clawfree graphs, Graphs Combin. 5 (009) [4] S. A. Choudum and M. S. Paulraj, Regular factors in K,3 free graphs, J. Graph Theory 5 (99) [5] Y. Egawa and K. Ota, Regular factors in K,n free graphs, J. Graph Theory 5 (99) [6] R. J. Faudree, O. Favaron, E. Flandrin, H. Li and Z. Liu, On factors in clawfree graphs, Discrete Math. 06 (999) [7] F. Harary and C. St. J. A. NashWilliams, On Eulerian and Hamiltonian graphs and line graphs, Can. Math. Bull. 8 (965) [8] B. Jackson and K. Yoshimoto, Even subgraphs of bridgeless graphs and factors of line graphs, Discrete Math. 307 (007) [9] B. Jackson and K. Yoshimoto, Spanning even subgraphs of 3edgeconnected graphs, J. Graph Theory 6 (009) [0] R. Kužel, K. Ozeki and K. Yoshimoto, factors and independent sets on clawfree graphs, Discrete Math. 3 (0), [] M. M. Matthews and D. P. Sumner, Hamiltonian results in K,3 free graphs, J. Graph Theory 8 (984) [] K. Ozeki, Z. Ryjáček and K. Yoshimoto, factors with bounded number of components in clawfree graphs, preprint. [3] Z. Ryjáček, On a closure concept in clawfree graphs, J. Combin. Theory Ser. B 70 (997) 7 4. [4] Z. Ryjáček, A. Saito, and R.H. Schelp, Closure, factors, and cycle coverings in clawfree graphs, J. Graph Theory 3 (999) [5] K. Yoshimoto, On the number of components in factors of clawfree graphs, Discrete Math. 307 (007)
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