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2-factors with the bounded number of components in line graphs
Applied Mathematics Letters 24 (2011) 731–734
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
2-factors with the bounded number of components in line graphs Liming Xiong ∗ Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China Department of Mathematics, Jiangxi Normal University, PR China Department of Mathematics, Qinghai University for Nationalities, PR China
article
info
Article history: Received 11 May 2009 Accepted 21 December 2010 Keywords: 2-factor Line graph Number of components Branch-bond
abstract Let G be a simple graph of order n such that every vertex of degree 1 is adjacent to a vertex of degree at least 3. In this work, we prove that the line graph L(G) has a 2-factor with at 1 most n− components if every odd branch-bond of G has a shortest branch of length 2. 3 This is a best possible result which can be thought of as a counterpart of the main result in Fujisawa et al. (2007) [8]. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction All graphs considered in this work are simple graphs. For notation and terminology not defined here, see [1]. A 2-factor of a graph G is a spanning subgraph such that every component is a cycle. In particular, a hamiltonian graph has a 2-factor with exactly one component. There are many results on the existence of 2-factors with a given number of components, mainly on the existence of hamiltonian graphs; see the survey paper [2]. Some examples showing the existence of 2-factors with number of components greater than 1 have appeared in [3–7]. Recently, Fujisawa et al. [8] studied the existence of 2-factors with a bounded number of components in line graphs; in this work we continue this direction of the research. A circuit is a connected graph with at least three vertices in which every vertex has even degree. A star is the complete bipartite graph K1,m . For a given graph G, we say that G has a k-system that dominates if there is a family S of edge-disjoint circuits and stars with at least three edges in G such that every edge of G is either in one of the circuits or stars, or is incident to a circuit in S , where k = |S |. The following result gives a characterization of the graph G such that its line graph L(G) contains a 2-factor with exactly k components. Theorem 1 (Gould and Hynds [9]). Let G be a graph without isolated vertices. The line graph L(G) contains a 2-factor with k (k ≥ 1) components if and only if G has a k-system that dominates. A graph is called trivial if it has only one vertex. A nontrivial path is called a branch if it has only internal vertices of degree 2 and end vertices of degree not 2. Note that a branch of length 1 has no internal vertex. The length of a branch is the number of its edges. We denote by B (G) the set of branches of G. For any subset S of B (G), we denote by G − S the subgraph obtained from G by deleting all edges and internal vertices of branches of S. A subset S of B (G) is called a branch cut if G − S has more components than G. A minimal branch cut is called a branch-bond. Obviously, for a connected graph G, a subset S of B (G) is a branch-bond if and only if G − S has exactly two components. A branch-bond is called odd if it has an odd
∗ Corresponding address: Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China. Tel.: +86 10 68912583; fax: +86 1082957305. E-mail addresses: [email protected], [email protected]. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.12.018
732
L. Xiong / Applied Mathematics Letters 24 (2011) 731–734
1 components. Fig. 1. A graph whose line graph has a 2-factor with n− 3
number of branches. The idea of a branch comes originally from [10] while that of a branch-bond is from [11], and has also been applied in [12]. The following result on the existence of a 2-factor with a bounded number of components involves the idea of branch-bonds, which extends a result of [13,14] stating that the line graph of a graph with minimum degree at least 3 has a 2-factor. Theorem 2 (Fujisawa et al. [8]). Let G be a simple graph of order n ≥ 4 with maximum degree at least 2. If every odd branch-bond of G has a shortest branch of length 1, then its line graph L(G) has a 2-factor with at most 3n8−2 components. There are many graphs G with an odd branch-bond containing a shortest branch of length 2 but their line graphs have no 2-factor even under the condition that G is 2-connected; see the examples in [8,12]. However, if every odd branch-bond of a graph has a shortest branch of length 2, then its line graph has a 2-factor. We obtain the following result which can be thought of as a counterpart of Theorem 2. Theorem 3. Let G be a simple graph of order n ≥ 4 such that every vertex of degree 1 has a neighbor of degree at least 3. If every 1 odd branch-bond of G has a shortest branch of length 2, then L(G) has a 2-factor with at most n− components. 3 The upper bound in Theorem 3 is best possible in the following sense. We use Z2 to denote the graph obtained from the disjoint union of a path P of length 2 and a triangle K3 by identifying one end vertex of P with one vertex of K3 . Let P4s = u0 u1 . . . u4s be a path of length 4s and Z200 , Z20 , Z21 , Z22 , . . . , Z2s , Z2s+1 s + 3 be copies of Z2 . Now obtain the graph G1 by identifying one end vertex of P4s with the two vertices of Z20 , Z200 of degree 1, respectively, and identifying the other end vertex of P4s with the two vertices of Z2s , Z2s+1 of degree 1, respectively, and identifying u4i with the vertex of Z2i of degree 1 for each i ∈ {1, 2, . . . , s − 1}, identifying u4i−2 with one end vertex of a disjoint additional edge for each i ∈ {1, 2, . . . , s}. For 1 the example in the case when s = 3, see Fig. 1. Then G1 has order n = 9s + 13 and n− = 3s + 4. Because each vertex of G1 3 1 with degree 3 must be in different circuits or stars in any possible k-system that dominates, G1 has the unique n− -system 3 1 that dominates. By Theorem 1, L(G1 ) has a 2-factor with exactly n− components. 3
Corollary 4. Let G be a simple graph of order n ≥ 4 with minimum degree at least 2. If every odd branch-bond of G has a shortest 1 branch of length 2, then L(G) has a 2-factor with at most n− components. 3 The upper bound in Corollary 4 is also best possible. The graph G2 obtained from G1 by contracting all stars with a pendent 1 edge has order m = 3s + 13 and a unique m− -system that dominates. 3 2. Auxiliary results We start with the following auxiliary result whose idea comes from [11]. The proof given here is similar to that of [8] which is a special case with ℓ = 1 of Theorem 5. For subgraphs H ⊂ F , let the interior IntF H = {u ∈ V (H ) | dF (u) ̸= 1}. Theorem 5. Let ℓ be a positive integer. Then a graph G has a set of vertex-disjoint circuits containing all branches of length not ℓ if every odd branch-bond in G has a shortest branch of length ℓ. Proof of Theorem 5. Let C1 , C2 , . . . , Ct be vertex-disjoint circuits in G such that C = i Ci contains branches of length not ℓ, as many as possible. Let F = G − V (C ), and suppose F contains a branch of G with length not ℓ, say P. Hence P has no end vertex of degree 1 in G. Since IntG (P ) ⊂ V (F ), E (P ) ⊂ E (G) \ E (C ). Let T be a maximal tree such that P ⊂ T and
ev ery edge of T ∩ C (if nonempty) is in a branch of G w ith length 1.
(2.1)
If we remove all edges and the internal vertices of P from T , then two trees T1 and T2 remain. Let B be a branch-bond of G joining T1 and G − V (T1 ) ∪ IntP (P ) in which P is one of the branches. We choose a branch B of G in B as follows. If B \ P has a branch of G which is edge-disjoint from C , then let B be the branch. In the case where B \ P has no such a branch, B is an odd branch-bond, and so B has a shortest branch of length ℓ. We choose B to be the shortest branch. Notice that if E (B) ∩ E (C ) ̸= ∅, then B is a branch of length ℓ. In either case, because of the maximality of T , B joins T1 and T2 , and so T ∪ B contains a cycle D. Then C ′ = ((C ∪ D) \ E (C ∩ D)) − IntC ∩D (C ∩ D) is a set of circuits. Because P ⊂ C ′ and any vertex in IntC ∩D (C ∩ D) is contained in a branch of length ℓ by (2.1), the set C ′ of the circuits contains more branches of length not ℓ than C , a contradiction. This completes the proof of Theorem 5.
L. Xiong / Applied Mathematics Letters 24 (2011) 731–734
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The following well-known result will be used in our proof. Theorem 6 (Veldman [15]). Let G be a graph with diameter at most 2. Then L(G) is hamiltonian, i.e., L(G) has a 2-factor with exactly one component. The following characterization of eulerian graphs involves branch-bonds. Theorem 7 (Xiong et al. [11]). A connected graph is eulerian if and only if every branch-bond contains an even number of branches. 3. Proof of the main result In this section we will present the proof of our main result. In the proofs we use the following notation and terminologies. Let Vi (H ) = {v ∈ V (H ) : dG (v) = i} for a subgraph H of G and ti (H ) = |Vi (H )|. A leaf of a tree is defined as a vertex of degree 1. An edge is called pendent if it has a vertex of degree 1. A branch of G is called an end-branch if it is a pendent edge. Proof of Theorem 3. Suppose that G has no odd branch-bond; then G is eulerian by Theorem 7 and hence L(G) is hamiltonian, and we are done. Now suppose that G has at least one odd branch-bond. Hence by the hypothesis of Theorem 3, G has at least one branch B0 of length 2. By Theorem 5, we can choose vertex-disjoint circuits C1 , C2 , . . . , Ck such that:
1. C = i≤k Ci contains all edges in those branches of length other than 2 that are not end-branches of G; 2. subject to 1, |V (C )| is maximal; 3. subject to the above, k is as small as possible. We claim that G − C is a forest. Suppose, to the contrary, that G − C has a cycle D0 . Then C ∪ D0 is an union of circuits which contains more vertices than C , contradicting requirement 2 for the choice of C . Suppose that there is no nontrivial component of G − C ; then either B0 is in some circuit Ci or the inner vertex of B0 has 1 and then we are done. degree 0 in G − C . In either case, since every circuit has at least three vertices, we have k ≤ n− 3 Now let T1 , T2 , . . . , Tf all be nontrivial components of G − C and F = i≥1 Ti . By requirement 1 for the choice of C , we have the following fact. Claim 1. T is not a path and any branch of T in which no end vertex has degree 1 has length exactly 2 for any T ∈ {T1 , T2 , . . . , Tf }. For each w ∈ W (F ) = {v ∈ i≥1 V (Ti ) : dF (v) ≥ 3}, let S (w) be the star {w u : u ∈ NF (w)}. By Claim 1, every edge of G is contained in C or w∈W (F ) S (w) or incident to C . Hence
{C1 , C2 , . . . , Ck } {S (w) : w ∈ W (F )} ∑ ∑ 1 is a (k + i≥3 ti (F ))-system that dominates. We will show that the number k + i≥3 ti (F ) is at most n− . 3 ∑ For each tree T ∈ {T1 , T2 , . . . , Tf }, we have |V (T )| = t ( T ) and | V ( T )| = | E ( T )| + 1. By Claim 1, |E (T )| = j ≥1 j 2t2 (T ) + t1 (T ). Therefore − tj (T ) = t2 (T ) + 1. (3.1) j≥3
So
|V (T )| =
−
tj (T ) = t1 (T ) + 2
j ≥1
−
ti (T ) − 1.
(3.2)
i≥3
On the other hand,
t1 (T ) + 2t2 (T ) + 3
−
ti (T ) ≤
v∈V (T )
i ≥3
implying t1 (T ) ≥
−
ti (T ) ≤
i ≥3
∑
i≥3 ti
−
dT (v) = 2|E (T )| = 2
−
ti (T ) − 1
i≥1
(T ) + 2. Combining this with (3.2), we obtain
|V (T )| − 1 3
.
Taking the sum for all trees Ti which satisfy the above inequality, we obtain
− i ≥3
ti (F ) ≤
|V (F )| − f 3
.
Because every circuit has at least three vertices, (3.3) and the assumption that f ≥ 1 imply
(3.3)
734
L. Xiong / Applied Mathematics Letters 24 (2011) 731–734
k+
−
ti (F ) ≤
i≥3
n − |V (F )| 3
+
|V (F )| − f
which finishes the proof of Theorem 3.
3
=
n−f 3
≤
n−1 3
,
Acknowledgements The research is supported by Natural Science Funds of China and by Beijing Natural Science Foundation (No. 1102015). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, American Elsevier, 1976. R. Gould, Advance on the hamiltonian problem—a survey, Graphs Combin. 19 (2003) 7–52. S. Brandt, G. Chen, R.J. Faudree, R.J. Gould, L. Lesniak, On the number of cycles in a 2-factor, J. Graph Theory 24 (1997) 165–173. R.J. Faudree, R.J. Gould, M.S. Jacobson, L. Lesniak, A. Saito, A note on 2-factors with two components, Discrete Math. 300 (2005) 218–224. R. Gould, E. Hynds, A note on 2-factors in line graphs, Bull. of ICA. 47 (2006) 58–62. R. Gould, M.S. Jacobson, Two-factors with few cycles in claw-free graphs, Discrete Math. 231 (2001) 191–197. A. Kaneko, K. Yoshimoto, On a 2-factor with a specified edge of a graph satisfying the Ore condition, Discrete Math. 257 (2002) 445–461. J. Fujisawa, L. Xiong, K. Yoshimoto, S. Zhang, The upper bound of the number of cycles in a 2-factor of a graph, J. Graph Theory 55 (2007) 72–82. R. Gould, E. Hynds, A note on cycles in 2-factor of line graphs, Bull. of ICA. 26 (1999) 46–48. P. Catlin, J. Iqbalunnisa, N. Srinivasan, Hamilton cycles and closed trails in iterated line graphs, J. Graph Theory 14 (1990) 347–364. L. Xiong, H.J. Broersma, X. Li, M. Li, The hamiltonian index of a graph and its branch-bonds, Discrete Math. 285 (2004) 279–288. L. Xiong, M. Li, On the 2-factor index of a graph, Discrete Math. 307 (2007) 2478–2483. S.A. Choudum, M.S. Paulraj, Regular factors in K1,3 -free graphs, J. Graph Theory 15 (1991) 259–265. Y. Egawa, K. Ota, Regular factors in K1,n -free graphs, J. Graph Theory 15 (1991) 337–344. H.J. Veldman, A result on hamiltonian line graphs involving restrictions on induced subgraphs, J. Graph Theory 12 (1988) 413–420.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
2-factors with the bounded number of components in line graphs Liming Xiong ∗ Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China Department of Mathematics, Jiangxi Normal University, PR China Department of Mathematics, Qinghai University for Nationalities, PR China
article
info
Article history: Received 11 May 2009 Accepted 21 December 2010 Keywords: 2-factor Line graph Number of components Branch-bond
abstract Let G be a simple graph of order n such that every vertex of degree 1 is adjacent to a vertex of degree at least 3. In this work, we prove that the line graph L(G) has a 2-factor with at 1 most n− components if every odd branch-bond of G has a shortest branch of length 2. 3 This is a best possible result which can be thought of as a counterpart of the main result in Fujisawa et al. (2007) [8]. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction All graphs considered in this work are simple graphs. For notation and terminology not defined here, see [1]. A 2-factor of a graph G is a spanning subgraph such that every component is a cycle. In particular, a hamiltonian graph has a 2-factor with exactly one component. There are many results on the existence of 2-factors with a given number of components, mainly on the existence of hamiltonian graphs; see the survey paper [2]. Some examples showing the existence of 2-factors with number of components greater than 1 have appeared in [3–7]. Recently, Fujisawa et al. [8] studied the existence of 2-factors with a bounded number of components in line graphs; in this work we continue this direction of the research. A circuit is a connected graph with at least three vertices in which every vertex has even degree. A star is the complete bipartite graph K1,m . For a given graph G, we say that G has a k-system that dominates if there is a family S of edge-disjoint circuits and stars with at least three edges in G such that every edge of G is either in one of the circuits or stars, or is incident to a circuit in S , where k = |S |. The following result gives a characterization of the graph G such that its line graph L(G) contains a 2-factor with exactly k components. Theorem 1 (Gould and Hynds [9]). Let G be a graph without isolated vertices. The line graph L(G) contains a 2-factor with k (k ≥ 1) components if and only if G has a k-system that dominates. A graph is called trivial if it has only one vertex. A nontrivial path is called a branch if it has only internal vertices of degree 2 and end vertices of degree not 2. Note that a branch of length 1 has no internal vertex. The length of a branch is the number of its edges. We denote by B (G) the set of branches of G. For any subset S of B (G), we denote by G − S the subgraph obtained from G by deleting all edges and internal vertices of branches of S. A subset S of B (G) is called a branch cut if G − S has more components than G. A minimal branch cut is called a branch-bond. Obviously, for a connected graph G, a subset S of B (G) is a branch-bond if and only if G − S has exactly two components. A branch-bond is called odd if it has an odd
∗ Corresponding address: Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China. Tel.: +86 10 68912583; fax: +86 1082957305. E-mail addresses: [email protected], [email protected]. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.12.018
732
L. Xiong / Applied Mathematics Letters 24 (2011) 731–734
1 components. Fig. 1. A graph whose line graph has a 2-factor with n− 3
number of branches. The idea of a branch comes originally from [10] while that of a branch-bond is from [11], and has also been applied in [12]. The following result on the existence of a 2-factor with a bounded number of components involves the idea of branch-bonds, which extends a result of [13,14] stating that the line graph of a graph with minimum degree at least 3 has a 2-factor. Theorem 2 (Fujisawa et al. [8]). Let G be a simple graph of order n ≥ 4 with maximum degree at least 2. If every odd branch-bond of G has a shortest branch of length 1, then its line graph L(G) has a 2-factor with at most 3n8−2 components. There are many graphs G with an odd branch-bond containing a shortest branch of length 2 but their line graphs have no 2-factor even under the condition that G is 2-connected; see the examples in [8,12]. However, if every odd branch-bond of a graph has a shortest branch of length 2, then its line graph has a 2-factor. We obtain the following result which can be thought of as a counterpart of Theorem 2. Theorem 3. Let G be a simple graph of order n ≥ 4 such that every vertex of degree 1 has a neighbor of degree at least 3. If every 1 odd branch-bond of G has a shortest branch of length 2, then L(G) has a 2-factor with at most n− components. 3 The upper bound in Theorem 3 is best possible in the following sense. We use Z2 to denote the graph obtained from the disjoint union of a path P of length 2 and a triangle K3 by identifying one end vertex of P with one vertex of K3 . Let P4s = u0 u1 . . . u4s be a path of length 4s and Z200 , Z20 , Z21 , Z22 , . . . , Z2s , Z2s+1 s + 3 be copies of Z2 . Now obtain the graph G1 by identifying one end vertex of P4s with the two vertices of Z20 , Z200 of degree 1, respectively, and identifying the other end vertex of P4s with the two vertices of Z2s , Z2s+1 of degree 1, respectively, and identifying u4i with the vertex of Z2i of degree 1 for each i ∈ {1, 2, . . . , s − 1}, identifying u4i−2 with one end vertex of a disjoint additional edge for each i ∈ {1, 2, . . . , s}. For 1 the example in the case when s = 3, see Fig. 1. Then G1 has order n = 9s + 13 and n− = 3s + 4. Because each vertex of G1 3 1 with degree 3 must be in different circuits or stars in any possible k-system that dominates, G1 has the unique n− -system 3 1 that dominates. By Theorem 1, L(G1 ) has a 2-factor with exactly n− components. 3
Corollary 4. Let G be a simple graph of order n ≥ 4 with minimum degree at least 2. If every odd branch-bond of G has a shortest 1 branch of length 2, then L(G) has a 2-factor with at most n− components. 3 The upper bound in Corollary 4 is also best possible. The graph G2 obtained from G1 by contracting all stars with a pendent 1 edge has order m = 3s + 13 and a unique m− -system that dominates. 3 2. Auxiliary results We start with the following auxiliary result whose idea comes from [11]. The proof given here is similar to that of [8] which is a special case with ℓ = 1 of Theorem 5. For subgraphs H ⊂ F , let the interior IntF H = {u ∈ V (H ) | dF (u) ̸= 1}. Theorem 5. Let ℓ be a positive integer. Then a graph G has a set of vertex-disjoint circuits containing all branches of length not ℓ if every odd branch-bond in G has a shortest branch of length ℓ. Proof of Theorem 5. Let C1 , C2 , . . . , Ct be vertex-disjoint circuits in G such that C = i Ci contains branches of length not ℓ, as many as possible. Let F = G − V (C ), and suppose F contains a branch of G with length not ℓ, say P. Hence P has no end vertex of degree 1 in G. Since IntG (P ) ⊂ V (F ), E (P ) ⊂ E (G) \ E (C ). Let T be a maximal tree such that P ⊂ T and
ev ery edge of T ∩ C (if nonempty) is in a branch of G w ith length 1.
(2.1)
If we remove all edges and the internal vertices of P from T , then two trees T1 and T2 remain. Let B be a branch-bond of G joining T1 and G − V (T1 ) ∪ IntP (P ) in which P is one of the branches. We choose a branch B of G in B as follows. If B \ P has a branch of G which is edge-disjoint from C , then let B be the branch. In the case where B \ P has no such a branch, B is an odd branch-bond, and so B has a shortest branch of length ℓ. We choose B to be the shortest branch. Notice that if E (B) ∩ E (C ) ̸= ∅, then B is a branch of length ℓ. In either case, because of the maximality of T , B joins T1 and T2 , and so T ∪ B contains a cycle D. Then C ′ = ((C ∪ D) \ E (C ∩ D)) − IntC ∩D (C ∩ D) is a set of circuits. Because P ⊂ C ′ and any vertex in IntC ∩D (C ∩ D) is contained in a branch of length ℓ by (2.1), the set C ′ of the circuits contains more branches of length not ℓ than C , a contradiction. This completes the proof of Theorem 5.
L. Xiong / Applied Mathematics Letters 24 (2011) 731–734
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The following well-known result will be used in our proof. Theorem 6 (Veldman [15]). Let G be a graph with diameter at most 2. Then L(G) is hamiltonian, i.e., L(G) has a 2-factor with exactly one component. The following characterization of eulerian graphs involves branch-bonds. Theorem 7 (Xiong et al. [11]). A connected graph is eulerian if and only if every branch-bond contains an even number of branches. 3. Proof of the main result In this section we will present the proof of our main result. In the proofs we use the following notation and terminologies. Let Vi (H ) = {v ∈ V (H ) : dG (v) = i} for a subgraph H of G and ti (H ) = |Vi (H )|. A leaf of a tree is defined as a vertex of degree 1. An edge is called pendent if it has a vertex of degree 1. A branch of G is called an end-branch if it is a pendent edge. Proof of Theorem 3. Suppose that G has no odd branch-bond; then G is eulerian by Theorem 7 and hence L(G) is hamiltonian, and we are done. Now suppose that G has at least one odd branch-bond. Hence by the hypothesis of Theorem 3, G has at least one branch B0 of length 2. By Theorem 5, we can choose vertex-disjoint circuits C1 , C2 , . . . , Ck such that:
1. C = i≤k Ci contains all edges in those branches of length other than 2 that are not end-branches of G; 2. subject to 1, |V (C )| is maximal; 3. subject to the above, k is as small as possible. We claim that G − C is a forest. Suppose, to the contrary, that G − C has a cycle D0 . Then C ∪ D0 is an union of circuits which contains more vertices than C , contradicting requirement 2 for the choice of C . Suppose that there is no nontrivial component of G − C ; then either B0 is in some circuit Ci or the inner vertex of B0 has 1 and then we are done. degree 0 in G − C . In either case, since every circuit has at least three vertices, we have k ≤ n− 3 Now let T1 , T2 , . . . , Tf all be nontrivial components of G − C and F = i≥1 Ti . By requirement 1 for the choice of C , we have the following fact. Claim 1. T is not a path and any branch of T in which no end vertex has degree 1 has length exactly 2 for any T ∈ {T1 , T2 , . . . , Tf }. For each w ∈ W (F ) = {v ∈ i≥1 V (Ti ) : dF (v) ≥ 3}, let S (w) be the star {w u : u ∈ NF (w)}. By Claim 1, every edge of G is contained in C or w∈W (F ) S (w) or incident to C . Hence
{C1 , C2 , . . . , Ck } {S (w) : w ∈ W (F )} ∑ ∑ 1 is a (k + i≥3 ti (F ))-system that dominates. We will show that the number k + i≥3 ti (F ) is at most n− . 3 ∑ For each tree T ∈ {T1 , T2 , . . . , Tf }, we have |V (T )| = t ( T ) and | V ( T )| = | E ( T )| + 1. By Claim 1, |E (T )| = j ≥1 j 2t2 (T ) + t1 (T ). Therefore − tj (T ) = t2 (T ) + 1. (3.1) j≥3
So
|V (T )| =
−
tj (T ) = t1 (T ) + 2
j ≥1
−
ti (T ) − 1.
(3.2)
i≥3
On the other hand,
t1 (T ) + 2t2 (T ) + 3
−
ti (T ) ≤
v∈V (T )
i ≥3
implying t1 (T ) ≥
−
ti (T ) ≤
i ≥3
∑
i≥3 ti
−
dT (v) = 2|E (T )| = 2
−
ti (T ) − 1
i≥1
(T ) + 2. Combining this with (3.2), we obtain
|V (T )| − 1 3
.
Taking the sum for all trees Ti which satisfy the above inequality, we obtain
− i ≥3
ti (F ) ≤
|V (F )| − f 3
.
Because every circuit has at least three vertices, (3.3) and the assumption that f ≥ 1 imply
(3.3)
734
L. Xiong / Applied Mathematics Letters 24 (2011) 731–734
k+
−
ti (F ) ≤
i≥3
n − |V (F )| 3
+
|V (F )| − f
which finishes the proof of Theorem 3.
3
=
n−f 3
≤
n−1 3
,
Acknowledgements The research is supported by Natural Science Funds of China and by Beijing Natural Science Foundation (No. 1102015). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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