- Email: [email protected]
On the difference between hamilton cycles and 2-factors with a prescribed number of cycles
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 61 (2017) 239–245 www.elsevier.com/locate/endm
On the difference between hamilton cycles and 2-factors with a prescribed number of cycles Shuya Chiba 1,2 Applied Mathematics Faculty of Advanced Science and Technology Kumamoto University 2-39-1 Kurokami, Kumamoto 860-8555, Japan
Abstract For a vertex subset X of a graph G, let Δ2 (X) be the maximum degree sum of two distinct vertices of X. In this paper, we give the following result: Let k be a positive integer, and let G be an m-connected graph of order n ≥ 5k − 2. If Δ2 (X) ≥ n for every independent set X of size m/k + 1 in G, then G has a 2-factor with exactly k cycles. This is a common generalization of the results obtained by Brandt et al. [Degree conditions for 2-factors, J. Graph Theory 24 (1997), 165–173] and Yamashita [On degree sum conditions for long cycles and cycles through specified vertices, Discrete Math. 308 (2008), 6584–6587], respectively. Keywords: Hamilton cycles, 2-factors, Vertex-disjoint cycles, Degree sum conditions
1 2
This work was supported by JSPS KAKENHI Grant Number 17K05347 Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2017.06.044 1571-0653/© 2017 Elsevier B.V. All rights reserved.
240
1
S. Chiba / Electronic Notes in Discrete Mathematics 61 (2017) 239–245
Introduction
In this paper, we consider finite simple graphs, which have neither loops nor multiple edges. For terminology and notation not defined in this paper, we refer the reader to [9]. The independence number and the connectivity of a graph G are denoted by α(G) and κ(G), respectively. For a vertex x of a graph G, we denote by dG (x) the degree of x in G. Let σm (G) be the minimum degree sum of an independent set of m vertices in a graph G, i.e., if α(G) ≥ m, then σm (G) = min dG (x) : X is an independent set of G with |X| = m ; x∈X
otherwise, σm (G) = +∞. If the graph G is clear from the context, we often omit the graph parameter G in the graph invariant. In this paper, “disjoint” always means “vertex-disjoint”. A graph having a hamilton cycle, i.e., a cycle containing all the vertices of the graph, is said to be hamiltonian. The hamiltonian problem has long been fundamental in graph theory. But, it is NP-complete, and so no easily verifiable necessary and sufficient condition seems to exist. Therefore, many researchers have focused on “better” sufficient conditions for graphs to be hamiltonian (see a survey [12]). In particular, the following degree sum condition, due to Ore (1960), is classical and well known. Theorem 1.1 (Ore [13]) Let G be a graph of order n ≥ 3. If σ2 ≥ n, then G is hamiltonian. Chv´atal and Erd˝os (1972) discovered the relationship between the connectivity, the independence number and the hamiltonicity. Theorem 1.2 (Chv´ atal and Erd˝ os [8]) Let G be a graph of order at least 3. If α ≤ κ, then G is hamiltonian. Bondy [2] pointed out that the graph satisfying the Ore condition also satisfies the Chv´atal-Erd˝os condition, that is, Theorem 1.2 implies Theorem 1.1. By Theorem 1.2, we should consider the degree condition for the existence of a hamilton cycle in graphs G with α(G) ≥ κ(G) + 1. In fact, Bondy (1980) gave the following degree sum condition by extending Theorem 1.2. Theorem 1.3 (Bondy [3]) Let G be an m-connected graph of order n ≥ 3. If σm+1 > 12 (m + 1)(n − 1), then G is hamiltonian.
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In 2008, Yamashita [15] introduced a new graph invariant and further generalized Theorem 1.3 as follows. For a vertex subset X of a graph G with |X| ≥ t, we define Δt (X) = max dG (x) : Y ⊆ X, |Y | = t . x∈Y
Let m ≥ t, and if α(G) ≥ m, then let m σt (G) = min Δt (X) : X is an independent set of G with |X| = m ; otherwise, σtm (G) = +∞. Note that σtm (G) ≥
t m
· σm (G).
Theorem 1.4 (Yamashita [15]) Let G be an m-connected graph of order n ≥ 3. If σ2m+1 ≥ n, then G is hamiltonian. This result suggests that the degree sum of non-adjacent “two” vertices is important for hamilton cycles. The purpose of this paper is to study the difference between hamilton cycles and 2-factors with a prescribed number of cycles in terms of degree sum conditions. In particular, in the next section, we give the Yamashita-type condition for 2-factors with exactly k cycles in order to generalize Theorem 1.4.
2
Degree sum conditions for 2-factors with a prescribed number of cycles
It is known that a 2-factor is one of the important generalizations of a hamilton cycle. A 2-factor is a spanning subgraph in which every component is a cycle, and thus a hamilton cycle is a 2-factor with “exactly 1 cycle”. As one of the studies concerning the difference between hamilton cycles and 2-factors, in this paper, we focus on 2-factors with “exactly k cycles”. Similar to the situation for hamilton cycles, deciding whether a graph has a 2-factor with k (≥ 2) cycles is also NP-complete. Therefore, the sufficient conditions for the existence of such a 2-factor have been extensively studied in graph theory (see a survey [11]). In particular, the following theorem, due to Brandt, Chen, Faudree, Gould and Lesniak (1997), is interesting. (In the paper [4], the order condition is not “n ≥ 4k − 1” but “n ≥ 4k”. However, by using a theorem of Enomoto [10] and Wang [14] (“every graph G of order at least 3k with σ2 (G) ≥ 4k − 1 contains k disjoint cycles”) for the cycles packing problem, we can obtain the following. See [4, Lemma 1].)
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Theorem 2.1 (Brandt et al. [4]) Let k be a positive integer, and let G be a graph of order n ≥ 4k − 1. If σ2 ≥ n, then G has a 2-factor with exactly k cycles. This theorem shows that the Ore condition guarantees the existence of a hamilton cycle but also the existence of a 2-factor with a prescribed number of cycles. By considering the relation between Theorem 1.1 and Theorem 2.1, Chen, Gould, Kawarabayashi, Ota, Saito and Schiermeyer [5] conjectured that the Chv´atal-Erd˝os condition in Theorem 1.2 also guarantees the existence of a 2-factor with exactly k cycles (see [5, Conjecture 1]). Chen et al. also proved that if the order of a 2-connected graph G with α(G) = α ≤ κ(G) is sufficiently large compared with k and with the Ramsey number r(α + 4, α + 1), then the graph G has a 2-factor with k cycles. Another related result can be found in [6]. But, the above conjecture is open in general. In this sense, there is a big gap between hamilton cycles and 2-factors with exactly k (≥ 2) cycles. In this paper, by combining the techniques of the proof for hamiltonicity and the proof for 2-factors with a prescribed number of cycles, we give the following Yamashita-type condition for 2-factors with k cycles. Theorem 2.2 Let k be a positive integer, and let G be an m-connected graph m/k+1 of order n ≥ 5k − 2. If σ2 ≥ n, then G has a 2-factor with exactly k cycles. Theorem 2.2 implies the following: Remark 2.3
•
Theorem 2.2 is a generalization of Theorem 1.4.
•
Theorem 2.2 leads to the Bondy-type condition: If G is an m-connected graph of order n ≥ 5k − 2 with σm/k+1 (G) > 12 (m/k + 1)(n − 1), then G has a 2-factor with exactly k cycles. Therefore, Theorem 2.2 is also a generalization of Theorem 2.1 for sufficiently large graphs. (Recall that σtm (G) ≥ mt · σm (G) and σm (G) ≥ m2 · σ2 (G) for m ≥ t ≥ 2.)
•
Theorem 2.2 leads to the Chv´atal-Erd˝os-type condition: If G is a graph of order at least 5k − 2 with α(G) ≤ κ(G)/k, then G has a 2-factor with exactly k cycles.
The complete bipartite graph K(n−1)/2,(n+1)/2 (n is odd) does not contain a 2-factor, and hence the degree condition in Theorem 2.2 is best possible in this sense. The order condition in Theorem 2.2 comes from our proof techniques. Similar to the situation for the proof of Theorem 2.1, we will use the order condition only for the cycles packing problem (for more details, we refer the
S. Chiba / Electronic Notes in Discrete Mathematics 61 (2017) 239–245
243
reader to our full version [7]). The complete bipartite graph K2k−1,2k−1 shows that n ≥ 4k − 1 is necessary. Table 1 summarizes the conditions mentioned in the above.
Ore-type
Chv´atal-Erd˝ os-type
Bondy-type
Yamashita-type
hamilton cycle
2-factor with k cycles
σ2 ≥ n
σ2 ≥ n
Theorem 1.1 (Ore)
Theorem 2.1 (Brandt et al.)
α≤κ
α ≤ κ/k
Theorem 1.2 (Chv´ atal and Erd˝ os)
Remark 2.3
σκ+1 > 12 (κ + 1)(n − 1)
σκ/k+1 > 12 (κ/k + 1)(n − 1)
Theorem 1.3 (Bondy)
Remark 2.3 κ/k+1
σ2κ+1 ≥ n Theorem 1.4 (Yamashita)
σ2
≥n
Theorem 2.2 (Main theorem)
Table 1 Comparison of the degree conditions
3
Outline of the proof of Theorem 2.2
In this section, we give the outline of the proof of Theorem 2.2. To prove Theorem 2.2, we extend the concept of insertible vertices which was introduced by Ainouche [1], and it plays a crucial role in our proof. Let G be a graph, and let D = {D1 , . . . , Ds } be the set of s disjoint cycles in G. For a vertex x in G − 1≤p≤s Dp , the vertex x is insertible for D if there is an edge uv in E(Dp ) such that xu, xv ∈ E(G) for some p with 1 ≤ p ≤ s. Then, we can obtain some lemmas which are useful tools for the proofs of the results on hamilton cycles and related topics as in the paper [1]. See our full paper [7, Lemmas 2–4] for more details. By using them, we can prove Theorem 2.2 as follows. Proof (Outline) Let C1 , . . . , Ck be k disjoint in G, and let C ∗ = cycles ∗ = 1≤p≤k |Ci | is as large as 1≤p≤k Cp . Choose C1 , . . . , Ck so that |C | ∗ possible. Suppose that C does not form a 2-factor of G. Let H = G − C ∗ ,
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and let H0 be a component of H and x0 ∈ V (H0 ). Let u1 , . . . , ul be l distinct vertices in NG (H0 ) ∩ V (C1 ), where l = m/k ≥ 2. We assume that u1 , u2 , . . . , ul appear in this order along the orientation of C1 . − Let u+ i and ui be the successor and the predecessor of ui along the orientation of C1 . Consider the set X = {x0 } ∪ {u+ i : 1 ≤ i ≤ l}. Using the crossing argument, we have the following: • •
•
X is an independent set of size l + 1. NG (x) ∩ V (H ∪ C1 ) + NG (x ) ∩ V (H ∪ C1 ) ≤ |H ∪ C1 | − 1 for x, x ∈ X with x = x . NG (x0 ) ∩ V (Cp ) ≤ |Cp |/2 for 2 ≤ p ≤ k. m/k+1
(G) ≥ n” and the defiBy combining this with the degree condition “σ2 nition of X, there is a vertex x ∈ X \ {x0 }, say x = u+ 1 , such that ∗ ∗ NG (u+ (1) 1 ) ∩ V (C − C1 ) > |C − C1 |/2. Next, we take the first non-insertible vertex x1 for {C2 , . . . , Ck } along the − reverse orientation of C1 in the path from u+ l to u1 on C1 , and we consider the set Y = {x0 , x1 } ∪ {u− i : 2 ≤ i ≤ l}. By using the concept of insertible vertices and the crossing argument, we have the following: • •
•
Y is an independent set of size l + 1. NG (y) ∩ V (H ∪ C1 ) + NG (y ) ∩ V (H ∪ C1 ) ≤ |H ∪ C1 | − 1 for y, y ∈ Y with y = y . NG (x1 ) ∩ V (Cp ) ≤ |Cp |/2 for 2 ≤ p ≤ k.
This, together with the degree condition and the definition of Y , implies that NG (u− ) ∩ V (C ∗ − C1 ) > |C ∗ − C1 |/2 for some i with 2 ≤ i ≤ l. (2) i Note that i ≥ 2, and hence (1) and (2) lead to the existence of k disjoint cycles in G such that the sum of the orders is larger than |C ∗ |, a contradiction. 2
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References [1] Ainouche, A., An improvement of Fraisse’s sufficient condition for hamiltonian graphs, J. Graph Theory 16 (1992), 529–543. [2] Bondy, J. A., A remark on two sufficient conditions for hamilton cycles, Discrete Math. 22 (1978), 191–193. [3] Bondy, J. A., “Longest paths and cycles in graphs with high degree,” Research Report CORR 80-16, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, 1980. [4] Brandt, S., G. Chen, R. Faudree, R. J. Gould and L. Lesniak, Degree conditions for 2-factors, J. Graph Theory 24 (1997), 165–173. [5] Chen, G., R. J. Gould, K. Kawarabayashi, K. Ota, A. Saito and I. Schiermeyer, The Chv´ atal-Erd˝ os condition and 2-factors with a specified number of components, Discuss. Math. Graph Theory 27 (2007), 401–407. [6] Chen, G., A. Saito and S. Shan, The existence of a 2-factor in a graph satisfying the local Chv´ atal-Erd˝ os condition, SIAM J. Discrete Math. 27 (2013), 1788– 1799. [7] Chiba, S., On degree sum conditions for 2-factors with a prescribed number of cycles, preprint. Available at arXiv:1705.02819. [8] Chv´atal, V., and P. Erd˝os, A note on hamiltonian circuits, Discrete Math. 2 (1972), 111–113. [9] Diestel, R., “Graph Theory,” 4th Ed., Graduate Texts in Mathematics, 173, Springer, Heidelberg, 2010. [10] Enomoto, H., On the existence of disjoint cycles in a graph, Combinatorica 18 (1998), 487–492. [11] Gould, R. J., Advances on the hamiltonian problem – a survey, Graphs and Combin. 19 (2003), 7–52. [12] Li, H., Generalizations of Dirac’s theorem in hamiltonian graph theory – a survey, Discrete Math. 313 (2013), 2034–2053. [13] Ore, O., Note on hamilton circuits, Amer. Math. Monthly 67 (1960), 55. [14] Wang, H., On the maximum number of independent cycles in a graph, Discrete Math. 205 (1999), 183–190. [15] Yamashita, T., On degree sum conditions for long cycles and cycles through specified vertices, Discrete Math. 308 (2008), 6584–6587.
Electronic Notes in Discrete Mathematics 61 (2017) 239–245 www.elsevier.com/locate/endm
On the difference between hamilton cycles and 2-factors with a prescribed number of cycles Shuya Chiba 1,2 Applied Mathematics Faculty of Advanced Science and Technology Kumamoto University 2-39-1 Kurokami, Kumamoto 860-8555, Japan
Abstract For a vertex subset X of a graph G, let Δ2 (X) be the maximum degree sum of two distinct vertices of X. In this paper, we give the following result: Let k be a positive integer, and let G be an m-connected graph of order n ≥ 5k − 2. If Δ2 (X) ≥ n for every independent set X of size m/k + 1 in G, then G has a 2-factor with exactly k cycles. This is a common generalization of the results obtained by Brandt et al. [Degree conditions for 2-factors, J. Graph Theory 24 (1997), 165–173] and Yamashita [On degree sum conditions for long cycles and cycles through specified vertices, Discrete Math. 308 (2008), 6584–6587], respectively. Keywords: Hamilton cycles, 2-factors, Vertex-disjoint cycles, Degree sum conditions
1 2
This work was supported by JSPS KAKENHI Grant Number 17K05347 Email: [email protected]
http://dx.doi.org/10.1016/j.endm.2017.06.044 1571-0653/© 2017 Elsevier B.V. All rights reserved.
240
1
S. Chiba / Electronic Notes in Discrete Mathematics 61 (2017) 239–245
Introduction
In this paper, we consider finite simple graphs, which have neither loops nor multiple edges. For terminology and notation not defined in this paper, we refer the reader to [9]. The independence number and the connectivity of a graph G are denoted by α(G) and κ(G), respectively. For a vertex x of a graph G, we denote by dG (x) the degree of x in G. Let σm (G) be the minimum degree sum of an independent set of m vertices in a graph G, i.e., if α(G) ≥ m, then σm (G) = min dG (x) : X is an independent set of G with |X| = m ; x∈X
otherwise, σm (G) = +∞. If the graph G is clear from the context, we often omit the graph parameter G in the graph invariant. In this paper, “disjoint” always means “vertex-disjoint”. A graph having a hamilton cycle, i.e., a cycle containing all the vertices of the graph, is said to be hamiltonian. The hamiltonian problem has long been fundamental in graph theory. But, it is NP-complete, and so no easily verifiable necessary and sufficient condition seems to exist. Therefore, many researchers have focused on “better” sufficient conditions for graphs to be hamiltonian (see a survey [12]). In particular, the following degree sum condition, due to Ore (1960), is classical and well known. Theorem 1.1 (Ore [13]) Let G be a graph of order n ≥ 3. If σ2 ≥ n, then G is hamiltonian. Chv´atal and Erd˝os (1972) discovered the relationship between the connectivity, the independence number and the hamiltonicity. Theorem 1.2 (Chv´ atal and Erd˝ os [8]) Let G be a graph of order at least 3. If α ≤ κ, then G is hamiltonian. Bondy [2] pointed out that the graph satisfying the Ore condition also satisfies the Chv´atal-Erd˝os condition, that is, Theorem 1.2 implies Theorem 1.1. By Theorem 1.2, we should consider the degree condition for the existence of a hamilton cycle in graphs G with α(G) ≥ κ(G) + 1. In fact, Bondy (1980) gave the following degree sum condition by extending Theorem 1.2. Theorem 1.3 (Bondy [3]) Let G be an m-connected graph of order n ≥ 3. If σm+1 > 12 (m + 1)(n − 1), then G is hamiltonian.
S. Chiba / Electronic Notes in Discrete Mathematics 61 (2017) 239–245
241
In 2008, Yamashita [15] introduced a new graph invariant and further generalized Theorem 1.3 as follows. For a vertex subset X of a graph G with |X| ≥ t, we define Δt (X) = max dG (x) : Y ⊆ X, |Y | = t . x∈Y
Let m ≥ t, and if α(G) ≥ m, then let m σt (G) = min Δt (X) : X is an independent set of G with |X| = m ; otherwise, σtm (G) = +∞. Note that σtm (G) ≥
t m
· σm (G).
Theorem 1.4 (Yamashita [15]) Let G be an m-connected graph of order n ≥ 3. If σ2m+1 ≥ n, then G is hamiltonian. This result suggests that the degree sum of non-adjacent “two” vertices is important for hamilton cycles. The purpose of this paper is to study the difference between hamilton cycles and 2-factors with a prescribed number of cycles in terms of degree sum conditions. In particular, in the next section, we give the Yamashita-type condition for 2-factors with exactly k cycles in order to generalize Theorem 1.4.
2
Degree sum conditions for 2-factors with a prescribed number of cycles
It is known that a 2-factor is one of the important generalizations of a hamilton cycle. A 2-factor is a spanning subgraph in which every component is a cycle, and thus a hamilton cycle is a 2-factor with “exactly 1 cycle”. As one of the studies concerning the difference between hamilton cycles and 2-factors, in this paper, we focus on 2-factors with “exactly k cycles”. Similar to the situation for hamilton cycles, deciding whether a graph has a 2-factor with k (≥ 2) cycles is also NP-complete. Therefore, the sufficient conditions for the existence of such a 2-factor have been extensively studied in graph theory (see a survey [11]). In particular, the following theorem, due to Brandt, Chen, Faudree, Gould and Lesniak (1997), is interesting. (In the paper [4], the order condition is not “n ≥ 4k − 1” but “n ≥ 4k”. However, by using a theorem of Enomoto [10] and Wang [14] (“every graph G of order at least 3k with σ2 (G) ≥ 4k − 1 contains k disjoint cycles”) for the cycles packing problem, we can obtain the following. See [4, Lemma 1].)
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Theorem 2.1 (Brandt et al. [4]) Let k be a positive integer, and let G be a graph of order n ≥ 4k − 1. If σ2 ≥ n, then G has a 2-factor with exactly k cycles. This theorem shows that the Ore condition guarantees the existence of a hamilton cycle but also the existence of a 2-factor with a prescribed number of cycles. By considering the relation between Theorem 1.1 and Theorem 2.1, Chen, Gould, Kawarabayashi, Ota, Saito and Schiermeyer [5] conjectured that the Chv´atal-Erd˝os condition in Theorem 1.2 also guarantees the existence of a 2-factor with exactly k cycles (see [5, Conjecture 1]). Chen et al. also proved that if the order of a 2-connected graph G with α(G) = α ≤ κ(G) is sufficiently large compared with k and with the Ramsey number r(α + 4, α + 1), then the graph G has a 2-factor with k cycles. Another related result can be found in [6]. But, the above conjecture is open in general. In this sense, there is a big gap between hamilton cycles and 2-factors with exactly k (≥ 2) cycles. In this paper, by combining the techniques of the proof for hamiltonicity and the proof for 2-factors with a prescribed number of cycles, we give the following Yamashita-type condition for 2-factors with k cycles. Theorem 2.2 Let k be a positive integer, and let G be an m-connected graph m/k+1 of order n ≥ 5k − 2. If σ2 ≥ n, then G has a 2-factor with exactly k cycles. Theorem 2.2 implies the following: Remark 2.3
•
Theorem 2.2 is a generalization of Theorem 1.4.
•
Theorem 2.2 leads to the Bondy-type condition: If G is an m-connected graph of order n ≥ 5k − 2 with σm/k+1 (G) > 12 (m/k + 1)(n − 1), then G has a 2-factor with exactly k cycles. Therefore, Theorem 2.2 is also a generalization of Theorem 2.1 for sufficiently large graphs. (Recall that σtm (G) ≥ mt · σm (G) and σm (G) ≥ m2 · σ2 (G) for m ≥ t ≥ 2.)
•
Theorem 2.2 leads to the Chv´atal-Erd˝os-type condition: If G is a graph of order at least 5k − 2 with α(G) ≤ κ(G)/k, then G has a 2-factor with exactly k cycles.
The complete bipartite graph K(n−1)/2,(n+1)/2 (n is odd) does not contain a 2-factor, and hence the degree condition in Theorem 2.2 is best possible in this sense. The order condition in Theorem 2.2 comes from our proof techniques. Similar to the situation for the proof of Theorem 2.1, we will use the order condition only for the cycles packing problem (for more details, we refer the
S. Chiba / Electronic Notes in Discrete Mathematics 61 (2017) 239–245
243
reader to our full version [7]). The complete bipartite graph K2k−1,2k−1 shows that n ≥ 4k − 1 is necessary. Table 1 summarizes the conditions mentioned in the above.
Ore-type
Chv´atal-Erd˝ os-type
Bondy-type
Yamashita-type
hamilton cycle
2-factor with k cycles
σ2 ≥ n
σ2 ≥ n
Theorem 1.1 (Ore)
Theorem 2.1 (Brandt et al.)
α≤κ
α ≤ κ/k
Theorem 1.2 (Chv´ atal and Erd˝ os)
Remark 2.3
σκ+1 > 12 (κ + 1)(n − 1)
σκ/k+1 > 12 (κ/k + 1)(n − 1)
Theorem 1.3 (Bondy)
Remark 2.3 κ/k+1
σ2κ+1 ≥ n Theorem 1.4 (Yamashita)
σ2
≥n
Theorem 2.2 (Main theorem)
Table 1 Comparison of the degree conditions
3
Outline of the proof of Theorem 2.2
In this section, we give the outline of the proof of Theorem 2.2. To prove Theorem 2.2, we extend the concept of insertible vertices which was introduced by Ainouche [1], and it plays a crucial role in our proof. Let G be a graph, and let D = {D1 , . . . , Ds } be the set of s disjoint cycles in G. For a vertex x in G − 1≤p≤s Dp , the vertex x is insertible for D if there is an edge uv in E(Dp ) such that xu, xv ∈ E(G) for some p with 1 ≤ p ≤ s. Then, we can obtain some lemmas which are useful tools for the proofs of the results on hamilton cycles and related topics as in the paper [1]. See our full paper [7, Lemmas 2–4] for more details. By using them, we can prove Theorem 2.2 as follows. Proof (Outline) Let C1 , . . . , Ck be k disjoint in G, and let C ∗ = cycles ∗ = 1≤p≤k |Ci | is as large as 1≤p≤k Cp . Choose C1 , . . . , Ck so that |C | ∗ possible. Suppose that C does not form a 2-factor of G. Let H = G − C ∗ ,
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S. Chiba / Electronic Notes in Discrete Mathematics 61 (2017) 239–245
and let H0 be a component of H and x0 ∈ V (H0 ). Let u1 , . . . , ul be l distinct vertices in NG (H0 ) ∩ V (C1 ), where l = m/k ≥ 2. We assume that u1 , u2 , . . . , ul appear in this order along the orientation of C1 . − Let u+ i and ui be the successor and the predecessor of ui along the orientation of C1 . Consider the set X = {x0 } ∪ {u+ i : 1 ≤ i ≤ l}. Using the crossing argument, we have the following: • •
•
X is an independent set of size l + 1. NG (x) ∩ V (H ∪ C1 ) + NG (x ) ∩ V (H ∪ C1 ) ≤ |H ∪ C1 | − 1 for x, x ∈ X with x = x . NG (x0 ) ∩ V (Cp ) ≤ |Cp |/2 for 2 ≤ p ≤ k. m/k+1
(G) ≥ n” and the defiBy combining this with the degree condition “σ2 nition of X, there is a vertex x ∈ X \ {x0 }, say x = u+ 1 , such that ∗ ∗ NG (u+ (1) 1 ) ∩ V (C − C1 ) > |C − C1 |/2. Next, we take the first non-insertible vertex x1 for {C2 , . . . , Ck } along the − reverse orientation of C1 in the path from u+ l to u1 on C1 , and we consider the set Y = {x0 , x1 } ∪ {u− i : 2 ≤ i ≤ l}. By using the concept of insertible vertices and the crossing argument, we have the following: • •
•
Y is an independent set of size l + 1. NG (y) ∩ V (H ∪ C1 ) + NG (y ) ∩ V (H ∪ C1 ) ≤ |H ∪ C1 | − 1 for y, y ∈ Y with y = y . NG (x1 ) ∩ V (Cp ) ≤ |Cp |/2 for 2 ≤ p ≤ k.
This, together with the degree condition and the definition of Y , implies that NG (u− ) ∩ V (C ∗ − C1 ) > |C ∗ − C1 |/2 for some i with 2 ≤ i ≤ l. (2) i Note that i ≥ 2, and hence (1) and (2) lead to the existence of k disjoint cycles in G such that the sum of the orders is larger than |C ∗ |, a contradiction. 2
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