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Forbidden pairs of disconnected graphs for traceability in connected graphs
Discrete Mathematics 340 (2017) 2194–2199
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Forbidden pairs of disconnected graphs for traceability in connected graphs Junfeng Du a , Binlong Li b , Liming Xiong a, * a b
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China Department of Applied Mathematics, Northwestern Polytechnical University, Xian, Shaanxi 710072, PR China
article
info
Article history: Received 22 September 2016 Received in revised form 14 February 2017 Accepted 24 April 2017
Keywords: Forbidden subgraph Traceability Disconnected graph
a b s t r a c t Let H be a class of given graphs. A graph G is said to be H-free if G contains no induced copies of H for every H ∈ H. A graph is called traceable if it contains a Hamilton path. Faudree and Gould (1997) characterized all the pairs {R, S } of connected subgraphs such that every connected {R, S }-free graph is traceable. Li and Vrána (2016) first consider the disconnected forbidden subgraphs, and characterized all pairs of disconnected forbidden subgraphs for hamiltonicity. In this article, we characterize all pairs {R, S } of graphs such that there exists an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable. © 2017 Elsevier B.V. All rights reserved.
1. Introduction We use Bondy and Murty [2] for terminology and notations not defined here and consider finite simple graphs only. Let G = (V (G), E(G)) be a graph. We use n(G), e(G), α (G), κ (G) and ω(G) to denote the order, size, independence number, connectivity and component number of G, respectively. Let u be a vertex of G. We use NG (u) to denote the set of vertices which is adjacent with u (also called the neighbors of u) in the graph G. Let S be a subset of V (G)(or E(G)). The induced subgraph of G is denoted by G[S ]. Furthermore, we use G − S to denote the subgraph G[V (G)\S ]. Let H be a given graph. A graph G is said to be H-free if G contains no induced copies of H. If G is H-free, then H is called a forbidden subgraph of G. For a class of graphs H, the graph G is H-free if G is H-free for every H ∈ H. Note that if H1 is an induced subgraph of H2 , then every H1 -free graph is also H2 -free. As usual, we use Kn to denote the complete graph of order n, and Km,n to denote the complete bipartite graph with partition sets of size m and n. So the K1 is a vertex, K3 is a triangle, K1,r is a star (the K1,3 is also called a claw). The clique C is a subset of vertices of a graph G such that G[C ] is a complete graph. Then we will show some special graphs which are needed: (see Fig. 1)
• • • •
Pi , the path with i vertices (note that P1 = K1 and P2 = K2 ); Zi , a graph obtained by identifying a vertex of a K3 with an end-vertex of a Pi+1 ; Li , a graph obtained by identifying a vertex of a Ki with an end-vertex of a P2 ; Ni,j,k , a graph obtained by identifying the three vertices of a K3 with an end-vertex of a Pi+1 , Pj+1 and Pk+1 , respectively.
A graph is called hamiltonian if it contains a Hamilton cycle, and is called traceable if it contains a Hamilton path. Bedrossian [1] characterized all pairs of connected forbidden subgraphs for hamiltonicity. Faudree and Gould [6] extended
*
Corresponding author. E-mail addresses: [email protected] (J. Du), [email protected] (B. Li), [email protected] (L. Xiong).
http://dx.doi.org/10.1016/j.disc.2017.04.017 0012-365X/© 2017 Elsevier B.V. All rights reserved.
J. Du et al. / Discrete Mathematics 340 (2017) 2194–2199
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Fig. 1. Some special graphs: Pi , Zi , Li and Ni,j,k .
Bedrossian’s result by proving the necessity part of the theorem based on infinite families of non-hamiltonian graphs, and characterized all the pairs {R, S } of connected subgraphs such that every connected {R, S }-free graph is traceable. Theorem 1 ([6]). Let R and S be connected graphs with R, S ̸ = P3 and let G be a connected graph. Then G being {R, S }-free implies G is traceable if and only if (up to symmetry) R = K1,3 and S is an induced subgraph of N1,1,1 . In [7], Li and Vrána first consider disconnected forbidden subgraphs, and characterized all pairs of disconnected forbidden subgraphs for hamiltonicity. Theorem 2 ([7]). Let S be a graph of order at least three. Then every 2-connected S-free graph is hamiltonian if and only if S is P3 or 3K1 . Theorem 3 ([7]). Let R and S be graphs of order at least three other than P3 and 3K1 . Then there is an integer n0 such that every 2-connected {R, S }-free graph of order at least n0 is hamiltonian, if and only if, one of the following is true (up to symmetry):
⋃ ⋃ ⋃ P4 ; Z1 , or K3 Z 2 , K2 • R = K1,3 and S is an induced subgraph of P6 , Z3 , N0,1,2 , N1⋃ , 1 , 1 , K1 • R = K1,k with k ≥ 4 and S is an induced subgraph of 2K1 K2 ; ⋃ • R = kK1 with k ≥ 4 and S is an induced subgraph of Ll with l ≥ 3, or 2K1 Kl′ with l′ ≥ 2. In this paper, we also consider disconnected forbidden subgraphs for traceability. First, we characterize all graphs S such that every connected S-free graph is traceable. Theorem 4. Let S be a graph of order at least three. Then every connected S-free graph is traceable if and only if S is P3 or 3K1 . Then we will characterize all pairs of disconnected {R, S } of graphs such that there exists an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable by the following theorems. Theorem 5. Let G be a connected K1,3 -free graph. Then (1) if G is also K1
⋃
(2) if G is also 2K1
Z1 -free and n(G) ≥ 13, then G is traceable;
⋃
K3 -free and n(G) ≥ 31, then G is traceable.
Theorem 6. Let G be a connected {K1,k , 2K1
⋃
K2 }-free graph, where k ≥ 4. If n(G) ≥ 2k + 1, then G is traceable.
The Ramsey number r(k, l) is defined as the smallest integer such that every graph of order r(k, l) contains either a clique of k vertices or an independent set of l vertices [9]. Theorem 7. Let G be a connected kK1 -free graph, where k ≥ 4. Then (1) if G is also Ll -free, n(G) ≥ r(2l − 3, k) + k − 2, then G is traceable; ⋃ where l ≥ 3, and (2) if G is also 2K1 Kl′ -free, where l′ ≥ 2, and n(G) ≥ r(2k + l′ − 2, k), then G is traceable. Finally, we may state our main result as follows. Theorem 8. Let R and S be graphs of order at least three other than P3 and 3K1 . Then there is an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable, if and only if, one of the following is true (up to symmetry):
⋃ ⋃ • R = K1,3 and S is an induced subgraph of N1,1,1 , K1 Z1 , ⋃ or 2K1 K3 ; • R = K1,k with k ≥ 4 and S is an induced subgraph of 2K1 K2 ; ⋃ • R = kK1 with k ≥ 4 and S is an induced subgraph of Ll with l ≥ 3, or 2K1 Kl′ with l′ ≥ 2. Note that there is a slight difference between Theorems 3 and 8 only in the first item.
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2. Preliminaries In this section, we will introduce some preliminary notations and theorems which are prepared for the proofs of Theorems 4 and 5. Let G be a given graph, the line graph of G, denoted by L(G), has E(G) as its vertex set, where two vertices in L(G) are adjacent if and only if the corresponding edges in G have at least one vertex in common. Ryjáček [10] introduced the line graph closure of a claw-free graph G, which becomes a useful tool in investigating traceable claw-free graphs. A vertex x ∈ V (G) is locally connected if the neighborhood of x induces a connected subgraph in G. For x ∈ V (G), the graph G′x obtained from G by adding the edges {yz : y, z ∈ N(x)andyz ̸ ∈ E(G)} is called the local completion of G at x. The closure of a claw-free graph G, denoted by cl(G), is obtained from G by recursively performing local completions at any locally connected vertex with non-complete neighborhood, as long as it is possible. A claw-free graph G is closed if G does not have any locally connected vertex with non-complete neighborhood. The following theorem translates claw-free graphs to line graphs when we consider the traceability of claw-free graphs. Theorem 9 (Ryjáček [10]). Let G be a claw-free graph. Then (1) cl(G) is uniquely determined; (2) cl(G) is claw-free; (3) cl(G) is the line graph of a triangle-free graph. Theorem 10 (Brandt, Favaron and Ryjáček [3]). G is traceable if and only if cl(G) is traceable. Following [4], we say a class H of graphs is stable if for every graph in H, its closure is also in H. Theorem ⋃ 11 (Brousek, Ryjáček and Favaron [4]). Let S be a connected claw-free closed graph of order at least 3. If S ∈ {K3 } i > 0} {Ni,j,k : i, j, k > 0}, then the class of {K1,3 , S }-free graphs is stable.
⋃ {Zi :
Theorem 12 ([7]). Let S be a disconnected claw-free closed graph. Then the class of {K1,3 , S }-free graphs is stable, if and only if for every component C of S, the class of {K1,3 , C }-free graphs is stable. K3 }-free graphs are stable. Z1 }-free graphs and the class of {K1,3 , 2K1 By Theorems 11 and 12, the class of {K1,3 , K1 For a closed claw-free graph G, we let L−1 (G) be a K3 -free graph such that G is its line graph. Clearly G is S-free if and only if L−1 (G) contains no (not necessary induced) copies of L−1 (S).
⋃
⋃
Theorem 13 ([8]). Every K3 -free graph of order n has at most n2 /4 edges. In 1972, Chvátal and Erdős [5] also gave the condition for hamiltonicity and traceability by α (G) and κ (G). Theorem 14 ([5]). For every graph G,
• if α (G) ≤ κ (G), then G is hamiltonian; • if α (G) ≤ κ (G) + 1, then G is traceable. 3. Proof of Theorem 4 We first construct some families of connected graphs Gi , i = 1, . . . , 6, that are not traceable (see Fig. 2). Let G1 = {G : G is obtained from a complete graph of order m1 , m1 ≥ 3, by adding three pendant edges to exactly three vertices in Km1 }; G2 = {G : G is obtained from a complete graph of order m2 , m2 ≥ 2, by identifying a vertex of K3 with one vertex of Km2 , and adding two pendant edges to exactly other two vertices in K3 }; G3 = {G : G is obtained from a complete graph of order m3 , m3 ≥ 2 by adding two pendant edges to exactly one vertex in Km3 }; G4 = {G : G is a star K1,m4 , m4 ≥ 3}; G5 = {G : G is N1,1,m5 , m5 ≥ 1}; G6 = {G : G is T1,1,m6 , m6 ≥ 1, which is obtained from a Pm6 +1 by adding two pendant edges to an end-vertex of Pm6 +1 }. The necessity part of Theorem 4. Let S be a graph of order at least three such that every connected S-free graph is traceable. Then S is an induced subgraph of all graphs in⋃ Gi , i = 1, . . . , 6. Note that the graphs in G1 do not have induced K1,3 ; the graphs in G3 only contain the subgraphs of 2K1 Kr⋃ , r ≥ 1 as disconnected induced subgraph, and do not have induced P4 ; the graphs in G4 do not have induced K3 and 2K1 Kr , r ≥ 2. Then if S is connected, S should be P3 , and if S is disconnected, S should be 3K1 . This completes the proof of the necessity part of Theorem 4. □ The sufficiency part of Theorem 4. If G is a connected P3 -free graph, then G is a complete graph, implying that G is traceable. If G is a connected 3K1 -free graph, then α (G) ≤ 2 ≤ κ (G) + 1, by Theorem 14, G is traceable. This completes the proof of the sufficiency part of Theorem 4. □
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Fig. 2. Some classes of graphs that are connected and non-traceable.
4. Proofs of Theorems 5 to 7 Proof of Theorem 5. Suppose G is a K1,3 -free non-traceable graph. By Theorems 9 and 10, it is sufficient to consider the case that G is closed. By Theorem 1, G contains an induced copy of N1,1,1 . Let H be a K3 -free graph such that H = L−1 (G). Hence H contains a (not necessary induced) copy of Q = L−1 (N1,1,1 ), that is a graph consisting of three paths Q1 = u∗ x0 y0 , Q2 = u∗ x1 y1 , Q3 = u∗ x2 y2 of length two, with the common original vertex u∗ . For convenience, we denote xi+1(mod3) (yi+1(mod3) ) and xi+2(mod3) (yi+2(mod3) ) by xi+1 (yi+1 ) and xi+2 (yi+2 ), respectively, i = 0, 1, 2. Case 1. G is {K1,3 , K1
⋃
Z1 }-free.
Claim 1. Every vertex in Q has no neighbor outside V (Q ). Z1 ), a contradiction. Proof. If u∗ has a neighbor u′ ∈ V (H) \ V (Q ), then H [{u∗ u′ , u∗ x0 , u∗ x1 , x0 y0 , x2 y2 }] is an L−1 (K1 ⋃ If Z1 ), xi , i = 0, 1, 2, has a neighbor x′i ∈ V (H) \ V (Q ), then the subgraph H [{xi x′i , u∗ xi , xi yi , u∗ xi+1 , xi+2 yi+2 }] is an L−1 (K1 ′ ′ ∗ ∗ ∗ a contradiction. If y , i = 0 , 1 , 2, has a neighbor y ∈ V (H) \ V (Q ), then H [{ u x , u x , u x , x y , y y }] is an i i i + 1 i + 2 i + 1 i + 1 i i i ⋃ L−1 (K1 Z1 ), a contradiction. □
⋃
By Claim ⋃ 1, n(H) = 7. By Theorem 13, since H is K3 -free, e(H) ≤ 12 and n(G) ≤ 12. This implies that every connected Z1 }-free graph of order at least 13 is traceable.
{K1,3 , K1
Case 2. G is {K1,3 , 2K1
⋃
K3 }-free.
Claim 2. Every vertex in Q has at most one neighbor outside V (Q ). Furthermore, xi , i = 0, 1, 2, have no neighbor outside V (Q ). ∗ Proof. If the vertex u⋃ has two neighbors u′ , u′′ ∈ V (H) \ V (Q ), then the subgraph H [{u∗ u′ , u∗ u′′ , u∗ x0 , −1 x1 y1 , x2 y2 }] is an L (2K1 K3 ), a contradiction. If⋃ xi ∈ {x0 , x1 , x2 } has a neighbor x′i ∈ V (H) \ V (Q ), then the subgraph H [{xi x′i , xi yi , u∗ xi , xi+1 yi+1 , xi+2 yi+2 }] is an L−1 (2K1 K3 ), a contradiction. If yi ∈ ⋃ {y0 , y1 , y2 } has two neighbors y′i , y′′i ∈ ′ ′′ −1 V (H) \ V (Q ), then the subgraph H [{xi yi , yi yi , yi yi , xi+1 yi+1 , xi+2 yi+2 }] is an L (2K1 K3 ), a contradiction. □
By Claim 2, we know the vertices u∗ , yi , i = 0, 1, 2, that may have a neighbor outside V (Q ). Since L(H [u∗ xi , i = 0, 1⋃ , 2]) is a K3 , at most one of the vertices {yi , i = 0, 1, 2} can have a neighbor outside V (Q ), otherwise we can find an L−1 (2K1 K3 ). Up to symmetry, we suppose that just yi has a neighbor outside V (Q ), called y′i . Then we can know that y′i can have at⋃most ⋃ a neighbor y′′i outside V (Q ) , and y′′i also has no neighbors outside V (Q ) {y′i }, otherwise we also can find an L−1 (2K1 K3 ). ′ ∗ ′ ′′ ′′ Then we consider the neighbor u of u outside V (Q ), u also can have at most a neighbor u outside V (Q ) and u also has no ⋃ neighbors outside V (Q ) {u′ }. It means⋃ that n(H) ≤ 11. By Theorem 13, since H is K3 -free, e(H) ≤ 30 and n(G) ≤ 30. This implies that every connected {K1,3 , 2K1 K3 }-free graph of order at least 31 is traceable. □ Proof of Theorem 6. Let G be a connected ⋃ {K1,k , S }-free graph, where S is an induced subgraph of 2K1 K2 . From [7], we know that every 2-connected {K1,k , 2K1 K2 }-free graph of order at least 2k is hamiltonian, hence we just need to consider the case when G has connectivity 1, i.e., G has a cut vertex u (say).
⋃
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Claim 3. Let G be a connected {K1,k , 2K1
⋃
K2 }-free graph, and let u be a cut vertex of G. If ω(G − u) ≥ 3, then |V (G)| ≤ k.
Proof. If ω(G − u) ≥ 3, then every component of G − u is trivial. Otherwise let e = x1 x2 be an edge from a nontrivial component of G − u, and let y1 , y2 be two vertices from two distinct components other than this component of G − u, ⋃ respectively. Then the subgraph G[{x1 , x2 , y1 , y2 }] is a copy of 2K1 K2 , a contradiction. Note that G is K1,k -free. This implies ω(G − u) < k, that is n(G) ≤ k. □ Claim 4. Let G be a connected {K1,k , 2K1 or G is traceable.
⋃
K2 }-free graph, and let u be a cut vertex of G. If ω(G − u) = 2, then either |V (G)| ≤ 2k
Proof. Let C1 , C2 be the two components ⋃ of G − u. First consider the case that |V (C1 )| ≥ 2 and |V (C2 )| ≥ 2. Since C1 , C2 are nontrivial components and G is 2K1 K2 -free, α (C1 ) < 2 and α (C2 ) < 2. This implies V (C1 ) and V (C2 ) both are cliques. It means G is traceable. Second, up to symmetry, suppose |V (C1 )| = 1. Let v ∈ V (C1 ). We consider the graph G − v , note that G − v is also a ⋃ connected {K1,k , 2K1 K2 }-free graph. In case when G − v is 2-connected, by [7], when n(G − v ) ≥ 2k, G − v is hamiltonian. It means that either n(G) ≤ 2k or G is traceable. Then we just need to consider the case when G − v also has a cut vertex x (say). By Claim 3, if ω(G − v − x) ≥ 3, then |V (G − v )| ≤ k. This implies that |V (G)| ≤ k + 1. Then we suppose that G − v − x also has two components C1′ ⋃ and C2′ . If |V (C1′ )| ≥ 2 and |V (C2′ )| ≥ 2, then C1′ , C2′ are nontrivial components. Therefore by the hypothesis that G is 2K1 K2 -free, α (C1′ ) < 2 and α (C2′ ) < 2. This implies that ′ ′ V (C1 ) and V (C2 ) both are cliques. It means that G − v and G both are traceable. Then, up to symmetry, we suppose that ′ |V (C1′ )| = 1 and V (C1′ ) = {y}. When y = u, it means that NG (v ) = {u} and NG−v (u) = {x}. Then ⋃ V (C2 ) must be a clique, otherwise the edge uv and two vertices in the independent set of C2′ form an induced copy of 2K1 K2 of G, a contradiction. This implies ⋃ that G is traceable. Then when y ̸ = u, it means that u ∈ V (C2′ ), NG (v ) = {u} and NG (y) = {x}. Since yv ̸ ∈ E(G) and G is 2K1 K2 -free, all edges of G must be incident ⋃ with x or u. Otherwise the edge which is not incident with x and u, and the vertices y, v form an induced copy of 2K1 K2 of G, a contradiction. Further, since G is K1,k -free, |NG (x)| < k and |NG (u)| < k. Since C2 , C2′ both are connected, n(G) ≤ k + 2 < 2k. □ By Claims 3 and 4, every connected {K1,k , 2K1
⋃
K2 }-free graph of order at least 2k + 1 is traceable. □
Kl′ . From Proof of Theorem 7. Let G be a connected {kK1 , S }-free graph, where S is an induced subgraph of Ll or 2K1 ⋃ [7], we Kl′ }-free know that every 2-connected {kK1 , Ll }-free graph of order at least r(2l − 3, k) + k − 2 and 2-connected {kK1 , 2K1 graph of order at least r(2k + l′ − 2, k) are hamiltonian. Then we just need to consider the case when G has connectivity 1, i.e., G has a cut vertex u (say).
⋃
Case 1. G is {kK1 , Ll }-free. Suppose that n(G) ≥ r(2l − 3, k) + k − 2 ≥ r(2l − 3, k) + 1. Then n(G − u) ≥ r(2l − 3, k). Since G and G − u are kK1 -free, G − u contains a clique C on 2l − 3 vertices. Because u is a cut vertex of G, the graph G − ⋂ u has at least two components. Let v be a vertex of G − u such that v and C are in distinct components of G − u. Then NG (v ) C = ⋂ ⋂ Ø. Let P′ = xx1 · · · xi v, i ≥ 1 ′ be a shortest path of G from v to C . Then V (P) V (C ) = { x } . Let C be the vertex set N (x ) C . If |C | ≥ l − 1, then the G 1 ⋃ subgraph G[C ′ {v}]⋃contains an Ll , a contradiction. If |C ′ | ≤ l − 2, then |C \ C ′ | ≥ 2l − 3 − (l − 2) = l − 1 and hence the subgraph G[(C \ C ′ ) {x, x1 }] contains an Ll , a contradiction. This implies that every {kK1 , Ll }-free graph of connectivity 1 and order at least r(2l − 3, k) + k − 2 is traceable. □ Case 2. G is {kK1 , 2K1 Kl′ }-free. ′ ′ Suppose ⋂ that n(G) ≥ r(2k + l − 2, k). Since G is kK1 -free, G contains a clique C on 2k + l − 2 vertices. Let X = {x ∈ V (G) : |NG (x) C | ≥ k} and Y = V (G) \ X . Then C ⊆ X (note that k ≥ 4).
⋃
Claim 5. Y is a clique. Proof. ⋂ Suppose, otherwise, that Y is not a clique. Then there are two vertices y1 , y2 ∈ Y such that y1 y2 ̸ ∈ E(G), and both |NG (yi ) C | ≤ k − 1, i = 1, 2. Then C has at least 2k +⋃l′ − 2 − 2(k − 1) = l′ vertices ⋃ that are nonadjacent to y1 and y2 . Let C ′ be the set of these vertices. Then the subgraph G[C ′ {y1 , y2 }] contains a 2K1 Kl′ , a contradiction. □ Since C is a clique and every vertex in X \ C has at least k neighbors in C , α (G[X ]) ≤ α (G) ≤ k ≤ κ (G[X ]). By Theorem 14, the subgraph G[X ] is hamiltonian. By Claim 5, the subgraph G[Y ] is hamiltonian. This implies that graph G has a hamiltonian path. ⋃ This implies that every {kK1 , 2K1 Kl′ }-free graph of connectivity 1 and order at least r(2k + l′ − 2, k) is traceable. □ 5. Proof of Theorem 8 By Theorems 1, 5, 6, 7, the sufficiency is proved. Then we just need to consider the necessity.
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Let R and S be graphs of order at least three other than P3 and 3K1 , such that there exists an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable. Then all graphs in Gi , i = 1, . . . , 6, of order large enough contain either R or S as an induced subgraph. Since the graphs in G4 are stars, an induced subgraph of order at least three of the graphs in G4 is a star or tK1 , t ≥ 3. Then either R or S is a star or tK1 , t ≥ 3. Thus we can assume without loss of generality that R is a star or tK1 , t ≥ 3. First we assume that R = K1,3 . Note that all graphs in G1 , G2 , G5 are K1,3 -free. Thus each of them contains S as an induced subgraph. Since all graphs in G2 are 4K1 -free, S is also 4K1 -free, i.e., α (S) ≤ 3. This implies that ω(S) ≤ 3. Suppose that ω(S) = 1, i.e., S is connected. Then, by Theorem 1, S is an induced subgraph of N1,1,1 . Suppose that ω(S) = 2, say, H1 , H2 are the two components of S. Since S is an induced subgraph of all graphs in G1 , either H1 or H2 must be K1 . We suppose without loss of generality that H1 = K1 . Since all graphs in G5 are⋃ Kt -free, t ≥ 4, and all graphs in G1 are Zt -free, t ≥ 2, H2 is an induced subgraph of Z1 . Thus S is an induced subgraph of K1 Z1 . Suppose that ω(S) = 3, say, H1 , H2 , H3 are the three components of S. Since S is an induced subgraph of all graphs in G1 , two of H1 , H2 and H3 must be K1 . We suppose ⋃ without loss of generality ⋃ that H1 = H2 = K1 . Then H3 is also an⋃induced subgraph of Z1 , but all graphs in G2 are 2K1 Z1 -free and contain 2K1 K3 , thus S is an induced subgraph of 2K1 K3 . Second we assume that R = K1,k with k ≥ 4. Note that all graphs⋃ in G1 , G2 , G⋃ 5 are also K1,4 -free. Thus each of them contains S as an induced subgraph. Then subgraph of K1 Z1 or 2K1 ⋃ K3 . Note that all graphs in G6 are also ⋃ S is an induced⋃ K1,4 -free, ⋃ and all graphs in G6 are K1 Z1 -free and 2K1 K3 -free, but contain 2K1 K2 . Thus S is an induced subgraph of K2 . 2K1 Finally we assume that R = kK1 with k ≥ 4. Note that all graphs in G2 , G3 are 4K1 -free. Thus each of them contains S as an induced subgraph. Then we can that S, the common induced subgraph of all graphs in G2 , G3 , is an induced ⋃ easily find Kl′ with l′ ≥ 2. This completes the proof of the necessity. □ subgraph of Ll with l ≥ 3 or 2K1 Concluding Remark. In this paper, we characterize all pairs {R, S } of graphs (not necessary connected) such that there exists an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable. Of course, for all induced subgraphs (not necessary connected) of R and S, these conclusions are still valid. In [7], Li and Vrána characterized all pairs of disconnected forbidden subgraphs for hamiltonicity in 2-connected graph. Most of our conclusions are the same as theirs, because of the relevance of traceability in connected graph and hamiltonicity in 2-connected graph. However, when R = K1,3 , our class of possible graphs S is strictly smaller than that of [7]. Acknowledgments This work is supported by the National Science Foundation of China (Nos. 11471037, 11671037 and 11601429). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Forbidden pairs of disconnected graphs for traceability in connected graphs Junfeng Du a , Binlong Li b , Liming Xiong a, * a b
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China Department of Applied Mathematics, Northwestern Polytechnical University, Xian, Shaanxi 710072, PR China
article
info
Article history: Received 22 September 2016 Received in revised form 14 February 2017 Accepted 24 April 2017
Keywords: Forbidden subgraph Traceability Disconnected graph
a b s t r a c t Let H be a class of given graphs. A graph G is said to be H-free if G contains no induced copies of H for every H ∈ H. A graph is called traceable if it contains a Hamilton path. Faudree and Gould (1997) characterized all the pairs {R, S } of connected subgraphs such that every connected {R, S }-free graph is traceable. Li and Vrána (2016) first consider the disconnected forbidden subgraphs, and characterized all pairs of disconnected forbidden subgraphs for hamiltonicity. In this article, we characterize all pairs {R, S } of graphs such that there exists an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable. © 2017 Elsevier B.V. All rights reserved.
1. Introduction We use Bondy and Murty [2] for terminology and notations not defined here and consider finite simple graphs only. Let G = (V (G), E(G)) be a graph. We use n(G), e(G), α (G), κ (G) and ω(G) to denote the order, size, independence number, connectivity and component number of G, respectively. Let u be a vertex of G. We use NG (u) to denote the set of vertices which is adjacent with u (also called the neighbors of u) in the graph G. Let S be a subset of V (G)(or E(G)). The induced subgraph of G is denoted by G[S ]. Furthermore, we use G − S to denote the subgraph G[V (G)\S ]. Let H be a given graph. A graph G is said to be H-free if G contains no induced copies of H. If G is H-free, then H is called a forbidden subgraph of G. For a class of graphs H, the graph G is H-free if G is H-free for every H ∈ H. Note that if H1 is an induced subgraph of H2 , then every H1 -free graph is also H2 -free. As usual, we use Kn to denote the complete graph of order n, and Km,n to denote the complete bipartite graph with partition sets of size m and n. So the K1 is a vertex, K3 is a triangle, K1,r is a star (the K1,3 is also called a claw). The clique C is a subset of vertices of a graph G such that G[C ] is a complete graph. Then we will show some special graphs which are needed: (see Fig. 1)
• • • •
Pi , the path with i vertices (note that P1 = K1 and P2 = K2 ); Zi , a graph obtained by identifying a vertex of a K3 with an end-vertex of a Pi+1 ; Li , a graph obtained by identifying a vertex of a Ki with an end-vertex of a P2 ; Ni,j,k , a graph obtained by identifying the three vertices of a K3 with an end-vertex of a Pi+1 , Pj+1 and Pk+1 , respectively.
A graph is called hamiltonian if it contains a Hamilton cycle, and is called traceable if it contains a Hamilton path. Bedrossian [1] characterized all pairs of connected forbidden subgraphs for hamiltonicity. Faudree and Gould [6] extended
*
Corresponding author. E-mail addresses: [email protected] (J. Du), [email protected] (B. Li), [email protected] (L. Xiong).
http://dx.doi.org/10.1016/j.disc.2017.04.017 0012-365X/© 2017 Elsevier B.V. All rights reserved.
J. Du et al. / Discrete Mathematics 340 (2017) 2194–2199
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Fig. 1. Some special graphs: Pi , Zi , Li and Ni,j,k .
Bedrossian’s result by proving the necessity part of the theorem based on infinite families of non-hamiltonian graphs, and characterized all the pairs {R, S } of connected subgraphs such that every connected {R, S }-free graph is traceable. Theorem 1 ([6]). Let R and S be connected graphs with R, S ̸ = P3 and let G be a connected graph. Then G being {R, S }-free implies G is traceable if and only if (up to symmetry) R = K1,3 and S is an induced subgraph of N1,1,1 . In [7], Li and Vrána first consider disconnected forbidden subgraphs, and characterized all pairs of disconnected forbidden subgraphs for hamiltonicity. Theorem 2 ([7]). Let S be a graph of order at least three. Then every 2-connected S-free graph is hamiltonian if and only if S is P3 or 3K1 . Theorem 3 ([7]). Let R and S be graphs of order at least three other than P3 and 3K1 . Then there is an integer n0 such that every 2-connected {R, S }-free graph of order at least n0 is hamiltonian, if and only if, one of the following is true (up to symmetry):
⋃ ⋃ ⋃ P4 ; Z1 , or K3 Z 2 , K2 • R = K1,3 and S is an induced subgraph of P6 , Z3 , N0,1,2 , N1⋃ , 1 , 1 , K1 • R = K1,k with k ≥ 4 and S is an induced subgraph of 2K1 K2 ; ⋃ • R = kK1 with k ≥ 4 and S is an induced subgraph of Ll with l ≥ 3, or 2K1 Kl′ with l′ ≥ 2. In this paper, we also consider disconnected forbidden subgraphs for traceability. First, we characterize all graphs S such that every connected S-free graph is traceable. Theorem 4. Let S be a graph of order at least three. Then every connected S-free graph is traceable if and only if S is P3 or 3K1 . Then we will characterize all pairs of disconnected {R, S } of graphs such that there exists an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable by the following theorems. Theorem 5. Let G be a connected K1,3 -free graph. Then (1) if G is also K1
⋃
(2) if G is also 2K1
Z1 -free and n(G) ≥ 13, then G is traceable;
⋃
K3 -free and n(G) ≥ 31, then G is traceable.
Theorem 6. Let G be a connected {K1,k , 2K1
⋃
K2 }-free graph, where k ≥ 4. If n(G) ≥ 2k + 1, then G is traceable.
The Ramsey number r(k, l) is defined as the smallest integer such that every graph of order r(k, l) contains either a clique of k vertices or an independent set of l vertices [9]. Theorem 7. Let G be a connected kK1 -free graph, where k ≥ 4. Then (1) if G is also Ll -free, n(G) ≥ r(2l − 3, k) + k − 2, then G is traceable; ⋃ where l ≥ 3, and (2) if G is also 2K1 Kl′ -free, where l′ ≥ 2, and n(G) ≥ r(2k + l′ − 2, k), then G is traceable. Finally, we may state our main result as follows. Theorem 8. Let R and S be graphs of order at least three other than P3 and 3K1 . Then there is an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable, if and only if, one of the following is true (up to symmetry):
⋃ ⋃ • R = K1,3 and S is an induced subgraph of N1,1,1 , K1 Z1 , ⋃ or 2K1 K3 ; • R = K1,k with k ≥ 4 and S is an induced subgraph of 2K1 K2 ; ⋃ • R = kK1 with k ≥ 4 and S is an induced subgraph of Ll with l ≥ 3, or 2K1 Kl′ with l′ ≥ 2. Note that there is a slight difference between Theorems 3 and 8 only in the first item.
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2. Preliminaries In this section, we will introduce some preliminary notations and theorems which are prepared for the proofs of Theorems 4 and 5. Let G be a given graph, the line graph of G, denoted by L(G), has E(G) as its vertex set, where two vertices in L(G) are adjacent if and only if the corresponding edges in G have at least one vertex in common. Ryjáček [10] introduced the line graph closure of a claw-free graph G, which becomes a useful tool in investigating traceable claw-free graphs. A vertex x ∈ V (G) is locally connected if the neighborhood of x induces a connected subgraph in G. For x ∈ V (G), the graph G′x obtained from G by adding the edges {yz : y, z ∈ N(x)andyz ̸ ∈ E(G)} is called the local completion of G at x. The closure of a claw-free graph G, denoted by cl(G), is obtained from G by recursively performing local completions at any locally connected vertex with non-complete neighborhood, as long as it is possible. A claw-free graph G is closed if G does not have any locally connected vertex with non-complete neighborhood. The following theorem translates claw-free graphs to line graphs when we consider the traceability of claw-free graphs. Theorem 9 (Ryjáček [10]). Let G be a claw-free graph. Then (1) cl(G) is uniquely determined; (2) cl(G) is claw-free; (3) cl(G) is the line graph of a triangle-free graph. Theorem 10 (Brandt, Favaron and Ryjáček [3]). G is traceable if and only if cl(G) is traceable. Following [4], we say a class H of graphs is stable if for every graph in H, its closure is also in H. Theorem ⋃ 11 (Brousek, Ryjáček and Favaron [4]). Let S be a connected claw-free closed graph of order at least 3. If S ∈ {K3 } i > 0} {Ni,j,k : i, j, k > 0}, then the class of {K1,3 , S }-free graphs is stable.
⋃ {Zi :
Theorem 12 ([7]). Let S be a disconnected claw-free closed graph. Then the class of {K1,3 , S }-free graphs is stable, if and only if for every component C of S, the class of {K1,3 , C }-free graphs is stable. K3 }-free graphs are stable. Z1 }-free graphs and the class of {K1,3 , 2K1 By Theorems 11 and 12, the class of {K1,3 , K1 For a closed claw-free graph G, we let L−1 (G) be a K3 -free graph such that G is its line graph. Clearly G is S-free if and only if L−1 (G) contains no (not necessary induced) copies of L−1 (S).
⋃
⋃
Theorem 13 ([8]). Every K3 -free graph of order n has at most n2 /4 edges. In 1972, Chvátal and Erdős [5] also gave the condition for hamiltonicity and traceability by α (G) and κ (G). Theorem 14 ([5]). For every graph G,
• if α (G) ≤ κ (G), then G is hamiltonian; • if α (G) ≤ κ (G) + 1, then G is traceable. 3. Proof of Theorem 4 We first construct some families of connected graphs Gi , i = 1, . . . , 6, that are not traceable (see Fig. 2). Let G1 = {G : G is obtained from a complete graph of order m1 , m1 ≥ 3, by adding three pendant edges to exactly three vertices in Km1 }; G2 = {G : G is obtained from a complete graph of order m2 , m2 ≥ 2, by identifying a vertex of K3 with one vertex of Km2 , and adding two pendant edges to exactly other two vertices in K3 }; G3 = {G : G is obtained from a complete graph of order m3 , m3 ≥ 2 by adding two pendant edges to exactly one vertex in Km3 }; G4 = {G : G is a star K1,m4 , m4 ≥ 3}; G5 = {G : G is N1,1,m5 , m5 ≥ 1}; G6 = {G : G is T1,1,m6 , m6 ≥ 1, which is obtained from a Pm6 +1 by adding two pendant edges to an end-vertex of Pm6 +1 }. The necessity part of Theorem 4. Let S be a graph of order at least three such that every connected S-free graph is traceable. Then S is an induced subgraph of all graphs in⋃ Gi , i = 1, . . . , 6. Note that the graphs in G1 do not have induced K1,3 ; the graphs in G3 only contain the subgraphs of 2K1 Kr⋃ , r ≥ 1 as disconnected induced subgraph, and do not have induced P4 ; the graphs in G4 do not have induced K3 and 2K1 Kr , r ≥ 2. Then if S is connected, S should be P3 , and if S is disconnected, S should be 3K1 . This completes the proof of the necessity part of Theorem 4. □ The sufficiency part of Theorem 4. If G is a connected P3 -free graph, then G is a complete graph, implying that G is traceable. If G is a connected 3K1 -free graph, then α (G) ≤ 2 ≤ κ (G) + 1, by Theorem 14, G is traceable. This completes the proof of the sufficiency part of Theorem 4. □
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Fig. 2. Some classes of graphs that are connected and non-traceable.
4. Proofs of Theorems 5 to 7 Proof of Theorem 5. Suppose G is a K1,3 -free non-traceable graph. By Theorems 9 and 10, it is sufficient to consider the case that G is closed. By Theorem 1, G contains an induced copy of N1,1,1 . Let H be a K3 -free graph such that H = L−1 (G). Hence H contains a (not necessary induced) copy of Q = L−1 (N1,1,1 ), that is a graph consisting of three paths Q1 = u∗ x0 y0 , Q2 = u∗ x1 y1 , Q3 = u∗ x2 y2 of length two, with the common original vertex u∗ . For convenience, we denote xi+1(mod3) (yi+1(mod3) ) and xi+2(mod3) (yi+2(mod3) ) by xi+1 (yi+1 ) and xi+2 (yi+2 ), respectively, i = 0, 1, 2. Case 1. G is {K1,3 , K1
⋃
Z1 }-free.
Claim 1. Every vertex in Q has no neighbor outside V (Q ). Z1 ), a contradiction. Proof. If u∗ has a neighbor u′ ∈ V (H) \ V (Q ), then H [{u∗ u′ , u∗ x0 , u∗ x1 , x0 y0 , x2 y2 }] is an L−1 (K1 ⋃ If Z1 ), xi , i = 0, 1, 2, has a neighbor x′i ∈ V (H) \ V (Q ), then the subgraph H [{xi x′i , u∗ xi , xi yi , u∗ xi+1 , xi+2 yi+2 }] is an L−1 (K1 ′ ′ ∗ ∗ ∗ a contradiction. If y , i = 0 , 1 , 2, has a neighbor y ∈ V (H) \ V (Q ), then H [{ u x , u x , u x , x y , y y }] is an i i i + 1 i + 2 i + 1 i + 1 i i i ⋃ L−1 (K1 Z1 ), a contradiction. □
⋃
By Claim ⋃ 1, n(H) = 7. By Theorem 13, since H is K3 -free, e(H) ≤ 12 and n(G) ≤ 12. This implies that every connected Z1 }-free graph of order at least 13 is traceable.
{K1,3 , K1
Case 2. G is {K1,3 , 2K1
⋃
K3 }-free.
Claim 2. Every vertex in Q has at most one neighbor outside V (Q ). Furthermore, xi , i = 0, 1, 2, have no neighbor outside V (Q ). ∗ Proof. If the vertex u⋃ has two neighbors u′ , u′′ ∈ V (H) \ V (Q ), then the subgraph H [{u∗ u′ , u∗ u′′ , u∗ x0 , −1 x1 y1 , x2 y2 }] is an L (2K1 K3 ), a contradiction. If⋃ xi ∈ {x0 , x1 , x2 } has a neighbor x′i ∈ V (H) \ V (Q ), then the subgraph H [{xi x′i , xi yi , u∗ xi , xi+1 yi+1 , xi+2 yi+2 }] is an L−1 (2K1 K3 ), a contradiction. If yi ∈ ⋃ {y0 , y1 , y2 } has two neighbors y′i , y′′i ∈ ′ ′′ −1 V (H) \ V (Q ), then the subgraph H [{xi yi , yi yi , yi yi , xi+1 yi+1 , xi+2 yi+2 }] is an L (2K1 K3 ), a contradiction. □
By Claim 2, we know the vertices u∗ , yi , i = 0, 1, 2, that may have a neighbor outside V (Q ). Since L(H [u∗ xi , i = 0, 1⋃ , 2]) is a K3 , at most one of the vertices {yi , i = 0, 1, 2} can have a neighbor outside V (Q ), otherwise we can find an L−1 (2K1 K3 ). Up to symmetry, we suppose that just yi has a neighbor outside V (Q ), called y′i . Then we can know that y′i can have at⋃most ⋃ a neighbor y′′i outside V (Q ) , and y′′i also has no neighbors outside V (Q ) {y′i }, otherwise we also can find an L−1 (2K1 K3 ). ′ ∗ ′ ′′ ′′ Then we consider the neighbor u of u outside V (Q ), u also can have at most a neighbor u outside V (Q ) and u also has no ⋃ neighbors outside V (Q ) {u′ }. It means⋃ that n(H) ≤ 11. By Theorem 13, since H is K3 -free, e(H) ≤ 30 and n(G) ≤ 30. This implies that every connected {K1,3 , 2K1 K3 }-free graph of order at least 31 is traceable. □ Proof of Theorem 6. Let G be a connected ⋃ {K1,k , S }-free graph, where S is an induced subgraph of 2K1 K2 . From [7], we know that every 2-connected {K1,k , 2K1 K2 }-free graph of order at least 2k is hamiltonian, hence we just need to consider the case when G has connectivity 1, i.e., G has a cut vertex u (say).
⋃
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Claim 3. Let G be a connected {K1,k , 2K1
⋃
K2 }-free graph, and let u be a cut vertex of G. If ω(G − u) ≥ 3, then |V (G)| ≤ k.
Proof. If ω(G − u) ≥ 3, then every component of G − u is trivial. Otherwise let e = x1 x2 be an edge from a nontrivial component of G − u, and let y1 , y2 be two vertices from two distinct components other than this component of G − u, ⋃ respectively. Then the subgraph G[{x1 , x2 , y1 , y2 }] is a copy of 2K1 K2 , a contradiction. Note that G is K1,k -free. This implies ω(G − u) < k, that is n(G) ≤ k. □ Claim 4. Let G be a connected {K1,k , 2K1 or G is traceable.
⋃
K2 }-free graph, and let u be a cut vertex of G. If ω(G − u) = 2, then either |V (G)| ≤ 2k
Proof. Let C1 , C2 be the two components ⋃ of G − u. First consider the case that |V (C1 )| ≥ 2 and |V (C2 )| ≥ 2. Since C1 , C2 are nontrivial components and G is 2K1 K2 -free, α (C1 ) < 2 and α (C2 ) < 2. This implies V (C1 ) and V (C2 ) both are cliques. It means G is traceable. Second, up to symmetry, suppose |V (C1 )| = 1. Let v ∈ V (C1 ). We consider the graph G − v , note that G − v is also a ⋃ connected {K1,k , 2K1 K2 }-free graph. In case when G − v is 2-connected, by [7], when n(G − v ) ≥ 2k, G − v is hamiltonian. It means that either n(G) ≤ 2k or G is traceable. Then we just need to consider the case when G − v also has a cut vertex x (say). By Claim 3, if ω(G − v − x) ≥ 3, then |V (G − v )| ≤ k. This implies that |V (G)| ≤ k + 1. Then we suppose that G − v − x also has two components C1′ ⋃ and C2′ . If |V (C1′ )| ≥ 2 and |V (C2′ )| ≥ 2, then C1′ , C2′ are nontrivial components. Therefore by the hypothesis that G is 2K1 K2 -free, α (C1′ ) < 2 and α (C2′ ) < 2. This implies that ′ ′ V (C1 ) and V (C2 ) both are cliques. It means that G − v and G both are traceable. Then, up to symmetry, we suppose that ′ |V (C1′ )| = 1 and V (C1′ ) = {y}. When y = u, it means that NG (v ) = {u} and NG−v (u) = {x}. Then ⋃ V (C2 ) must be a clique, otherwise the edge uv and two vertices in the independent set of C2′ form an induced copy of 2K1 K2 of G, a contradiction. This implies ⋃ that G is traceable. Then when y ̸ = u, it means that u ∈ V (C2′ ), NG (v ) = {u} and NG (y) = {x}. Since yv ̸ ∈ E(G) and G is 2K1 K2 -free, all edges of G must be incident ⋃ with x or u. Otherwise the edge which is not incident with x and u, and the vertices y, v form an induced copy of 2K1 K2 of G, a contradiction. Further, since G is K1,k -free, |NG (x)| < k and |NG (u)| < k. Since C2 , C2′ both are connected, n(G) ≤ k + 2 < 2k. □ By Claims 3 and 4, every connected {K1,k , 2K1
⋃
K2 }-free graph of order at least 2k + 1 is traceable. □
Kl′ . From Proof of Theorem 7. Let G be a connected {kK1 , S }-free graph, where S is an induced subgraph of Ll or 2K1 ⋃ [7], we Kl′ }-free know that every 2-connected {kK1 , Ll }-free graph of order at least r(2l − 3, k) + k − 2 and 2-connected {kK1 , 2K1 graph of order at least r(2k + l′ − 2, k) are hamiltonian. Then we just need to consider the case when G has connectivity 1, i.e., G has a cut vertex u (say).
⋃
Case 1. G is {kK1 , Ll }-free. Suppose that n(G) ≥ r(2l − 3, k) + k − 2 ≥ r(2l − 3, k) + 1. Then n(G − u) ≥ r(2l − 3, k). Since G and G − u are kK1 -free, G − u contains a clique C on 2l − 3 vertices. Because u is a cut vertex of G, the graph G − ⋂ u has at least two components. Let v be a vertex of G − u such that v and C are in distinct components of G − u. Then NG (v ) C = ⋂ ⋂ Ø. Let P′ = xx1 · · · xi v, i ≥ 1 ′ be a shortest path of G from v to C . Then V (P) V (C ) = { x } . Let C be the vertex set N (x ) C . If |C | ≥ l − 1, then the G 1 ⋃ subgraph G[C ′ {v}]⋃contains an Ll , a contradiction. If |C ′ | ≤ l − 2, then |C \ C ′ | ≥ 2l − 3 − (l − 2) = l − 1 and hence the subgraph G[(C \ C ′ ) {x, x1 }] contains an Ll , a contradiction. This implies that every {kK1 , Ll }-free graph of connectivity 1 and order at least r(2l − 3, k) + k − 2 is traceable. □ Case 2. G is {kK1 , 2K1 Kl′ }-free. ′ ′ Suppose ⋂ that n(G) ≥ r(2k + l − 2, k). Since G is kK1 -free, G contains a clique C on 2k + l − 2 vertices. Let X = {x ∈ V (G) : |NG (x) C | ≥ k} and Y = V (G) \ X . Then C ⊆ X (note that k ≥ 4).
⋃
Claim 5. Y is a clique. Proof. ⋂ Suppose, otherwise, that Y is not a clique. Then there are two vertices y1 , y2 ∈ Y such that y1 y2 ̸ ∈ E(G), and both |NG (yi ) C | ≤ k − 1, i = 1, 2. Then C has at least 2k +⋃l′ − 2 − 2(k − 1) = l′ vertices ⋃ that are nonadjacent to y1 and y2 . Let C ′ be the set of these vertices. Then the subgraph G[C ′ {y1 , y2 }] contains a 2K1 Kl′ , a contradiction. □ Since C is a clique and every vertex in X \ C has at least k neighbors in C , α (G[X ]) ≤ α (G) ≤ k ≤ κ (G[X ]). By Theorem 14, the subgraph G[X ] is hamiltonian. By Claim 5, the subgraph G[Y ] is hamiltonian. This implies that graph G has a hamiltonian path. ⋃ This implies that every {kK1 , 2K1 Kl′ }-free graph of connectivity 1 and order at least r(2k + l′ − 2, k) is traceable. □ 5. Proof of Theorem 8 By Theorems 1, 5, 6, 7, the sufficiency is proved. Then we just need to consider the necessity.
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Let R and S be graphs of order at least three other than P3 and 3K1 , such that there exists an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable. Then all graphs in Gi , i = 1, . . . , 6, of order large enough contain either R or S as an induced subgraph. Since the graphs in G4 are stars, an induced subgraph of order at least three of the graphs in G4 is a star or tK1 , t ≥ 3. Then either R or S is a star or tK1 , t ≥ 3. Thus we can assume without loss of generality that R is a star or tK1 , t ≥ 3. First we assume that R = K1,3 . Note that all graphs in G1 , G2 , G5 are K1,3 -free. Thus each of them contains S as an induced subgraph. Since all graphs in G2 are 4K1 -free, S is also 4K1 -free, i.e., α (S) ≤ 3. This implies that ω(S) ≤ 3. Suppose that ω(S) = 1, i.e., S is connected. Then, by Theorem 1, S is an induced subgraph of N1,1,1 . Suppose that ω(S) = 2, say, H1 , H2 are the two components of S. Since S is an induced subgraph of all graphs in G1 , either H1 or H2 must be K1 . We suppose without loss of generality that H1 = K1 . Since all graphs in G5 are⋃ Kt -free, t ≥ 4, and all graphs in G1 are Zt -free, t ≥ 2, H2 is an induced subgraph of Z1 . Thus S is an induced subgraph of K1 Z1 . Suppose that ω(S) = 3, say, H1 , H2 , H3 are the three components of S. Since S is an induced subgraph of all graphs in G1 , two of H1 , H2 and H3 must be K1 . We suppose ⋃ without loss of generality ⋃ that H1 = H2 = K1 . Then H3 is also an⋃induced subgraph of Z1 , but all graphs in G2 are 2K1 Z1 -free and contain 2K1 K3 , thus S is an induced subgraph of 2K1 K3 . Second we assume that R = K1,k with k ≥ 4. Note that all graphs⋃ in G1 , G2 , G⋃ 5 are also K1,4 -free. Thus each of them contains S as an induced subgraph. Then subgraph of K1 Z1 or 2K1 ⋃ K3 . Note that all graphs in G6 are also ⋃ S is an induced⋃ K1,4 -free, ⋃ and all graphs in G6 are K1 Z1 -free and 2K1 K3 -free, but contain 2K1 K2 . Thus S is an induced subgraph of K2 . 2K1 Finally we assume that R = kK1 with k ≥ 4. Note that all graphs in G2 , G3 are 4K1 -free. Thus each of them contains S as an induced subgraph. Then we can that S, the common induced subgraph of all graphs in G2 , G3 , is an induced ⋃ easily find Kl′ with l′ ≥ 2. This completes the proof of the necessity. □ subgraph of Ll with l ≥ 3 or 2K1 Concluding Remark. In this paper, we characterize all pairs {R, S } of graphs (not necessary connected) such that there exists an integer n0 such that every connected {R, S }-free graph of order at least n0 is traceable. Of course, for all induced subgraphs (not necessary connected) of R and S, these conclusions are still valid. In [7], Li and Vrána characterized all pairs of disconnected forbidden subgraphs for hamiltonicity in 2-connected graph. Most of our conclusions are the same as theirs, because of the relevance of traceability in connected graph and hamiltonicity in 2-connected graph. However, when R = K1,3 , our class of possible graphs S is strictly smaller than that of [7]. Acknowledgments This work is supported by the National Science Foundation of China (Nos. 11471037, 11671037 and 11601429). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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