Let G be a graph on n ≥ 3 vertices. A graph G is almost distance-hereditary if each connected induced subgraph H of G has the property d_{H}(x, y) ≤ d_{G}(x, y) + 1 for any pair of vertices x, y ∈ V(H). Adopting the terminology introduced by Broersma et al. and Čada, a graph G is called 1-heavy if at least one of the end vertices of each induced subgraph of G isomorphic to K_{1,3} (a claw) has degree at least n/2, and is called claw-heavy if each claw of G has a pair of end vertices with degree sum at least n. In this paper we prove the following two theorems: (1) Every 2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian. (2) Every 3-connected, 1-heavy and almost distance-hereditary graph is Hamiltonian. The first result improves a previous theorem of Feng and Guo [J.-F. Feng and Y.-B. Guo, Hamiltonian cycle in almost distance-hereditary graphs with degree condition restricted to claws, Optimazation57 (2008), no. 1, 135–141]. For the second result, its connectedness condition is sharp since Feng and Guo constructed a 2-connected 1-heavy graph which is almost distance-hereditary but not Hamiltonian.
In this paper, we only consider the graphs which are finite, undirected and without multi-edges and loops. For terminology and notation not defined here, we refer to Bondy and Murty [1].
Let G be a graph with vertex set V(G) and H be a subgraph of G. For a vertex υ ∈ V(G), we denote by N_{H}(υ) the set of vertices which are adjacent to υ in H, and by d_{H}(υ) = |N_{H}(υ)| the degree of υ in H. For two vertices x, y ∈ V(G), an (x, y)-path in H is a path starting from x to y with all vertices in H. The distance of x and y in H, denoted by d_{H}(x, y), is defined as the length of a shortest (x, y)-path in H. When there is no danger of ambiguity, we use N(υ), d(υ) and d(x, y) instead of N_{G}(υ), d_{G}(υ) and d_{G}(x, y), respectively.
A graph is called Hamiltonian if it contains a Hamilton cycle, i.e., a cycle passing through all the vertices of the graph. The study of cycles, especially Hamilton cycles, may be one of the most important and most studied areas of graph theory. It is well-known that to determine whether a given graph contains a Hamilton cycle is
A graph G is called almost distance-hereditary if each connected induced subgraph H of G has the property d_{H}(x, y) ≤ d_{G}(x, y) + 1 for any pair of vertices x, y ∈ V(H). For some properties and a characterization of almost-distance hereditary graphs, we refer to [7].
Let G be a graph. An induced subgraph of G isomorphic to K_{1,3} is called a claw. The vertex of degree 3 in the claw is called its center and the other vertices are its end vertices. G is called claw-free if G contains no claw. Throughout this paper, whenever the vertices of a claw are listed, its center is always the first one.
Many results about the existence of Hamilton cycles in claw-free graphs have been obtained. In particular, Feng and Guo [8] gave the following result on Hamiltonicity of almost distance-hereditary claw-free graphs.
(Feng and Guo [8]). Let G be a 2-connected claw-free graph. If G is almost distance-hereditary, then G is Hamiltonian.
Let G be a graph on n vertices. A vertex υ of G is called heavy if d(υ) ≥ n/2. Broersma et al. [9] introduced the concepts of 1-heavy graph and 2-heavy graph. Later, Fujisawa and Yamashita [10] and Čada [11] introduced the concept of claw-heavy graphs, independently. Following [9–11], we say that a claw in G is 1-heavy (2-heavy) if at least one (two) of its end vertices is (are) heavy. G is called 1-heavy (2-heavy) if every claw of it is 1-heavy (2-heavy), and called claw-heavy if every claw of it has two end vertices with degree sum at least n. It is easily seen that every claw-free graph is 1-heavy (2-heavy, claw-heavy), every 2-heavy graph is claw-heavy and every claw-heavy graph is 1-heavy. But not every claw-heavy graph is 2-heavy, and not every 1-heavy graph is claw-heavy.
In [12], Feng and Guo extended Theorem 1.1 to a larger graph class of 2-heavy graphs.
(Feng and Guo [12]). Let G be a 2-connected 2-heavy graph. If G is almost distance-hereditary, then G is Hamiltonian.
Feng and Guo [12] also constructed a 2-connected 1-heavy graph which is almost distance-hereditary but not Hamiltonian. Thus it is natural to ask which is the minimum connectivity for a 1-heavy almost distance-hereditary graph under this connectivity condition to be Hamiltonian.
Motivated by [9, 13–15], in this paper we obtain the following two theorems which extend Theorem 1.1 and Theorem 1.2. In particular, Theorem 1.3 improves Theorem 1.2, and Theorem 1.4 answers the problem proposed above.
Let G be a 2-connected claw-heavy graph. If G is almost distance-hereditary, then G is Hamiltonian.
Let G be a 3-connected 1-heavy graph. If G is almost distance-hereditary, then G is Hamiltonian.
We emphasize that our technique of proofs is different from Feng and Guo [12]. One of our main tools is the so called "Ore-cycle" (motivated by Lemma 3 in [13]) introduced by Li et al. [16].
The graph in Fig. 1 shows that the result in Theorem 1.3 indeed strengthen that in Theorem 1.2. As shown in [13, Fig. 2], let n ≥ 10 be an even integer and K_{n/2} + K_{n/2−3}denote the join of two complete graphs K_{n/2}and K_{n/2−3}. Choose a vertex y ∈ V(K_{n/2}) and construct a graph G with V(G) = V(K_{n/2} + K_{n/2−3}) ∪ {υ, u, x} and E(G) = E(K_{n/2} + K_{n/2−3}) ∪ {uυ, uy, ux} ∪ {υw, xw : w ∈ V(K_{n/2−3})}. It is easy to see that G is a Hamiltonian graph satisfying the condition of Theorem 1.3, but not the condition of Theorem 1.2.
We postpone the proofs of Theorem 1.3 and 1.4 to Section 3.
Let G be a graph on n vertices and k ≥ 3 be an integer. Recall that a vertex of degree at least n/2 in G is a heavy vertex; otherwise it is light. A claw in G is called a light claw if all its end vertices are light, and is called an o-light claw if any pair of end vertices has degree sum less than n. A cycle in G is called a heavy cycle if it contains all heavy vertices of G. Following [16], we use Ẽ(G) to denote the set {uυ : uυ ∈ E(G) or d(u) + d(υ) ≥ n, u, υ ∈ V(G)}. A sequence of vertices C = υ_{1}υ_{2} … υ_{k} υ_{1} is called an Ore-cycle or briefly, o-cycle of G, if we have υ_{i}υ_{i+1} ∈ Ẽ(G) for every i ∈ {1, 2, · , k}, where υ_{1} = υ_{k+1}.
Let G be a graph and k be a nonnegative integer. For a cycle C of G and a vertex u ∈ V(G)\V(C), a subgraph F of G is said to be a (u, C ; k)-fanif F is a union of paths P_{1}, P_{2}, … , P_{k}, where P_{i} is a (u, w_{i})-path (1 ≤ i ≤ k), P_{i} ∩ C = {w_{i}} (1 ≤ i ≤ k) and P_{i} ∩ P_{j} = {u} for 1 ≤ i < j ≤ k. In the following, we use F = (u: P_{1}, P_{2}, … , P_{k}) to denote the fan. The vertices in V(F)\{w_{1}, w_{2}, … , w_{k}} are called internal vertices of F.
We need some notations from [13]. Let H be a path or a cycle with a given orientation. We denote by
To prove Theorems 1.3 and 1.4, the following six lemmas are needed. In particular, similar proofs of the facts in Lemma 2.3 can be found in [13, 16] (for example, see Claims 1-4 of Theorem 8 in [16]). For the convenience of the readers, we write the detailed proofs here.
(Li, Wang, Ryjáček, Zhang [16]). Let G be graph and let C′ be an o-cycle of G. Then there exists a cycle C of G such that V(C′) ⊆ V(C).
Let G be a non-Hamiltonian graph on n vertices, C be a longest cycle (a longest heavy cycle) of G, R a component of G − V(C), and A = {υ_{1}, υ_{2}, … , υ_{3}} the set of neighbors of R on C. Let u ∈ V(R) andυ_{i}, υ_{j} ∈ A. Then there hold
(a) uυ^{−}_{i} ∉ Ẽ(G), uυ^{+}_{i} ∉ Ẽ(G); (b) υ^{−}_{i}υ^{−}_{j} ∉ Ẽ(G), υ^{+}_{i}υ^{+}_{j} ∉ Ẽ(G); and (c)ifυ^{−}_{i}υ^{+}_{i} ∉ Ẽ(G), thenυ_{i}υ^{−}_{j} ∉ Ẽ(G), υ_{i}υ^{+}_{j} ∉ Ẽ(G).
Furthermore, if G is a 2-connected claw-heavy graph, then
(d) υ^{−}_{i}υ^{+}_{i} ∈ Ẽ(G) andυ^{−}_{j}υ^{+}_{j} ∈ Ẽ(G); (e) υ^{−}_{i}υ^{+}_{i} ∈ E(G) orυ^{−}_{j}υ^{+}_{j} ∈ E(G).
Proof. (a) Suppose that uυ^{−}_{i} ∈ Ẽ(G). Then C′ = uυ_{i}C[υ_{i}, υ^{−}_{i}]υ^{−}_{i}u is an o-cycle of length longer than C, a contradiction. The other assertion can be proved similarly.
(b) Suppose that υ^{−}_{i}υ^{−}_{j} ∈ Ẽ(G). Then
(c) Suppose that υ_{i}υ^{−}_{j} ∈ Ẽ(G). Then C′ = υ_{i}uυ_{j}C[υ_{j}, υ^{−}_{i}]υ^{−}_{i}υ^{+}_{i}C[υ^{+}_{i}, υ^{−}_{i}]υ^{−}_{i}υ_{i} is an o-cycle of length longer than C, a contradiction. The other assertion can be proved similarly.
(d) If υ^{−}_{i}υ^{+}_{i} ∉ E(G), then by (a), we know uυ^{−}_{i} ∉ E(G), uυ^{+}_{i} ∉ E(G). Thus {υ_{i}, u, υ_{i}^{−}, υ_{i}^{+}}, induces a claw. Since G is claw-heavy, by (a), we have d(υ^{−}_{i}) + d(υ^{+}_{i}) ≥ n. This implies that υ^{−}_{i}υ^{+}_{i} ∈ Ẽ(G). If υ^{−}_{i}υ^{+}_{i} ∈ E(G), then obviously υ^{−}_{i}υ^{+}_{i} ∈ Ẽ(G). The other assertion can be proved similarly.
(e) Suppose that υ^{−}_{i}υ^{+}_{i} ∉ E(G) and υ^{−}_{j}v^{+}_{j} ∉ E(G). By (d), we have d(υ^{−}_{i}) + d(υ^{+}_{i}) ≥ n and d(υ^{−}_{j}) + d(υ^{+}_{j}) ≥ n. This implies that d(υ^{−}_{i}) + d(υ^{−}_{j}) ≥ n or d(υ^{+}_{i}) + d(υ^{+}_{j}) ≥ n. Thus υ^{−}_{i}υ^{−}_{j} ∈ Ẽ(G) or υ^{+}_{i}υ^{+}_{j} ∈ Ẽ(G), a contradiction to (b).
Let G be a non-Hamiltonian graph, C be a longest cycle (a longest heavy cycle) of G, R a component of G − V(C), and A = {υ_{1}, υ_{2}, . . . , υ_{k}} the set of neighbors of R on C. Let υ_{i}, υ_{j} ∈ A. Then there hold
(a) forl ∈ V(C(υ_{i}, υ^{−}_{j}]), ifυ^{−}_{i}l ∈ Ẽ(G), thenl^{−}υ^{+}_{j} ∉ Ẽ(G) andl^{+}υ^{+}_{j} ∉ Ẽ(G);
(b) forl ∈ V(C(υ_{i}, υ^{−}_{j}]) ∩ N(υ_{i}), ifυ^{−}_{i}υ^{+}_{i} ∈ Ẽ(G), thenl^{−}υ^{−}_{j} ∉ Ẽ(G) andl^{+}υ^{+}_{j} ∉ Ẽ(G); and
(c) forl ∈ V(C(υ_{i}, υ^{−}_{j}]) ∩ N(υ_{i}) ∩ N(υ^{−}_{j}), ifυ^{−}_{i}υ^{+}_{i} ∈ Ẽ(G), thenl^{−}l^{+} ∉ Ẽ(G).
Proof. Let P be a (υ_{i}, υ_{j})-path with all internal vertices in R.
(a) Suppose l^{−}υ^{+}_{j} ∈ Ẽ(G). Then
(b) Suppose l^{−}υ^{−}_{j} ∈ Ẽ(G). Then
(c) Suppose l^{−}l^{+} ∈ Ẽ(G). Then C′ = PC[υ_{j}, υ^{−}_{i}]υ^{−}_{i}υ^{+}_{i}C[υ^{+}_{i}, l^{−}]l^{−}l^{+}C[l^{+}, υ^{−}_{j}]υ^{−}_{j}lυ_{i} is an o-cycle such that V(C) ⊂ V(C′), a contradiction. □
Let G be a 3-connected 1-heavy non-Hamiltonian graph and C be a longest heavy cycle of G, R a component of G − V(C). Let u ∈ V(R), and υ_{0}, υ_{1}, υ_{2}be three neighbors of u which are in the order around C andυ^{−}_{1}, υ^{+}_{1}are light. Letl_{i} ∈ C[υ^{+}_{i}, υ^{−}_{i+1}) such thatυ^{−}_{i+1}l_{i} ∈ E(G) and l_{i} υ_{i} ∈ E(G), s_{i} ∈ C(υ^{+}_{i−1}, υ^{−}_{i}] such thatυ^{+}_{i−1}s_{i} ∈ E(G) and s_{i} υ_{i} ∈ E(G) (where the indices are taken modulo 3).
(a) Ifυ^{−}_{2}υ^{+}_{2} ∉ E(G) andυ^{−}_{0}υ^{+}_{0} ∉ E(G), then (i) υ^{−}_{1}υ_{2} ∉ E(G), (ii) υ_{1}l^{−}_{1} ∈ E(G) andυ_{1}s^{+}_{1} ∈ E(G), (iii) υ_{1}, l^{+}_{1}, l^{−}_{1}, s^{−}_{1}, s^{+}_{1}, l_{1}, s_{1}are light;
(b) Ifυ^{−}_{2}υ^{+}_{2} ∈ E(G) andυ^{−}_{0}υ^{+}_{0} ∈ E(G), then {υ^{+}_{1}, l^{+}_{0}, s^{+}_{2}} induces an independent set.
Proof. By Lemma 2.3 (a), uυ^{−}_{1} ∉ E(G) and uυ^{+}_{1} ∉ E(G). If υ^{−}_{1}υ^{+}_{1} ∉ E(G), then since u, υ^{−}_{1}, υ^{+}_{1} are light, {υ_{1}, u, υ^{−}_{1}, υ^{+}_{1}} induces a light claw, a contradiction. Thus υ^{−}_{1}υ^{+}_{1} ∈ E(G).
(a) Since υ^{−}_{2}υ^{+}_{2} ∉ E(G) and υ^{−}_{0}υ^{+}_{0} ∉ E(G), we have υ^{−}_{2}, υ^{+}_{0} are heavy or υ^{+}_{2}, υ^{−}_{0} are heavy by Lemma 2.3 (b).
(i) Suppose υ^{−}_{1}υ_{2} ∈ E(G) and υ^{−}_{2}, υ^{+}_{0} are heavy. Let C′ = υ^{−}_{1}υ_{2}C[υ_{2}, υ_{0}]υ_{2}uυ_{1}C[υ_{1}, υ^{−}_{2}]υ^{−}_{2}υ^{+}_{0}C[υ^{+}_{0}, υ^{−}_{1}]. Then C′ is an o-cycle such that V(C) ⊂ V(C′), a contradiction.
Suppose υ^{−}_{1}υ_{2} ∈ E(G) and υ^{+}_{2}, υ^{−}_{0} are heavy. Now {υ_{2}, υ^{−}_{2}, u, υ^{−}_{1}} induces a light claw, a contradiction.
(ii) Suppose υ_{1}l^{−}_{1} ∉ E(G). Note that υ_{1}l^{+}_{1} ∉ Ẽ(G) and l^{−}_{1}l^{+}_{1} ∉ Ẽ(G) by Lemma 2.4 (b) and (c). Since υ^{−}_{2}, υ^{+}_{0} are heavy or υ^{+}_{2}, υ^{−}_{0} are heavy, by Lemma 2.3 (c) and Lemma 2.4 (b), υ_{1}, l^{+}_{1}, l^{−}_{1} are light. Now {l_{1}, l^{+}_{1}, υ_{1}, l^{−}_{1}} induces a light claw, a contradiction. Similarly, we can prove that υ_{1}s^{+}_{1} ∈ E(G).
(iii) By Lemma 2.3 (c) and Lemma 2.4 (b), υ_{1}, l^{+}_{1}, l^{−}_{1}, s^{−}_{1}, s^{+}_{1} are light. Since υ_{1}l^{−}_{1} ∈ E(G), we obtain υ^{+}_{2}l_{1} ∉ Ẽ(G) and υ^{+}_{0}l_{1} ∉ Ẽ(G) by Lemma 2.4 (b). Note that either υ^{+}_{0} or υ^{+}_{2} is a heavy vertex. This implies l_{1} is a light vertex. The other assertion that s_{1} is light can be proved similarly.
(b) Since υ_{0}l_{0} ∈ E(G) and υ_{2}s_{2} ∈ E(G), υ^{+}_{1}l^{+}_{0} ∉ Ẽ(G) and υ^{+}_{1}s^{+}_{2} ∈ Ẽ(G) by Lemma 2.4 (b). Furthermore, we can prove that l^{+}_{0}s^{+}_{2} ∉ E(G). (Otherwise,
Let G be a non-Hamiltonian almost distance-hereditary graph, C be a longest cycle (a longest heavy cycle) of G, R a component of G − V(C). If there exists a vertex u ∈ V(R) such that N_{C}(u) = {υ_{1}; υ_{2}, … , υ_{r}}, then there hold
(a) for any induced (u, υ)-path P, whereυ ∈ N^{−}_{C}(u) orυ ∈ N^{+}_{C}(u), the length of P is at most 3;
(b) ifυ^{−}_{i}υ^{+}_{i} ∈ Ẽ(G), then there exists a vertexl_{i} ∈ C[υ^{+}_{i}, υ^{−}_{i+1}) such thatυ^{−}_{i+1}l_{i} ∈ E(G) and l_{i}υ_{i} ∈ E(G), and there exists a vertexs_{i} ∈ C(υ^{+}_{i−1}, υ^{−}_{1}] such thatυ^{+}_{i−1}s_{i} ∈ E(G) and s_{i} υ_{i} ∈ E(G);
(c) ifυ^{−}_{i}υ^{+}_{i} ∈ E(G) andυ^{−}_{i+1}υ^{+}_{i+1} ∈ E(G), thenυ^{+}_{i+1}l_{i} ∈ E(G); and
(d) ififυ^{−}_{i}υ^{+}_{i} ∈ E(G) andυ^{−}_{i+1}υ^{+}_{i+1} ∈ E(G), then both {l_{i}, l^{−}_{i}, υ_{i}, υ^{−}_{i+1}} and {l_{i}, l^{+}_{i}, υ_{i}, υ^{+}_{i+1}} induce claws.
Proof. (a) Suppose there exists an induced (u, υ)-path P such that the length of P is at least 4. It follows that d_{P} (u, υ) ≥ 4, contradicting the fact that d_{G}(u, υ) = 2 and G is almost distance-hereditary.
(b) Let H = G[{u} ∪ V(C[υ_{i}, υ_{i+1}])] − {υ_{i+1}}. Since d_{G}(υ^{−}_{i+1}, u) = 2 and G is almost distance-hereditary, we have d_{H}(υ^{−}_{i+1}, u) ≤ 3. Since υ^{−}_{i}υ^{+}_{i} ∈ Ẽ(G), by Lemma 2.3 (c), we have υ_{i}υ^{−}_{i+1} ∉ E(G) and d_{H}(υ^{−}_{i+1}, u) = 3. It follows that d_{H}(υ^{−}_{i+1}, υ_{i}) = 2. So there exists a vertex l_{i} ∈ C[υ^{+}_{i}, υ^{−}_{i+1}) such that υ^{−}_{i+1}l_{i} ∈ E(G) and l_{i} υ_{i} ∈ E(G). The other assertion can be proved similarly.
(c) Suppose υ^{+}_{i+1}l_{i} ∉ E(G). Let H = G[{υ^{+}_{i+1}, υ^{−}_{i+1}, l_{i}, υ_{i}, u}]. By Lemma 2.3 (c), υ_{i}υ^{−}_{i+1} ∉ E(G) and υ_{i}υ^{+}_{i+1} ∉ E(G). We can see H is an induced (u, υ^{+}_{i+1})-path of length 4 in G, contradicting Lemma 2.6 (a).
(d) By Lemma 2.3 (c) and Lemma 2.4 (b), we have υ_{i}υ^{−}_{i+1} ∉ Ẽ(G) and l^{−}_{i}υ^{−}_{i+1} ∉ Ẽ(G). By Lemma 2.6 (c) and Lemma 2.4 (b), we have υ_{i}l^{−}_{i} ∉ E(G). So {l_{i}, l^{−}_{i}, υ_{i}, υ^{−}_{i+1}} induces a claw. The other assertion can be proved similarly. □
Proof of Theorem 1.3
Let G be a graph satisfying the condition of Theorem 1.3. Let C be a longest cycle of G and assign an orientation to it. Suppose G is not Hamiltonian. Then
|V(C(υ_{i}, υ_{j}))| is as small as possible;
|V(P)| is as small as possible subject to (1).
There is no o-cycle C′ in G such that V(C) ⊂ V(C′).
Proof. Otherwise, C′ is an o-cycle such that V(C) ⊂ V(C′). By Lemma 2.2, there exists a cycle containing all vertices in C′ and longer than C, contradicting the choice of C. □
By Lemma 2.3 (e), without loss of generality, assume that υ^{−}_{i}υ^{+}_{i} ∈ E(G).
r = 1, that is, V(P) = {υ_{j}, u_{1}, υ_{j}}.
Proof. Suppose r ≥ 2. Consider H = G[V(P) ∪ V(C[υ_{i}, υ_{j}])] − {υ_{j}}. Since υ^{−}_{i}υ^{+}_{i} ∈ E(G), υ_{i}υ^{−}_{j} ∉ E(G) by Lemma 2.3 (c). Thus d_{H}(υ^{−}_{j}, υ_{i}) ≥ 2. By the choice condition of P and Lemma 2.3 (a), we have d_{P}(υ_{i}, u_{r}) ≥ 2 and d_{H}(υ^{−}_{j}, u_{r}) = d_{H}(υ^{−}_{j}, υ_{i}) + d_{H}(υ_{j}, u_{r}) ≥ 4, which yields a contradiction to the fact G is almost distance-hereditary and d_{G}(υ^{−}_{j}, u_{r}) = 2. Hence V(P) = {υ_{i}, u_{1}, υ_{j}}. □
|V(C[υ_{i}, υ_{j}])| ≥ 5.
Proof. Suppose |V(C[υ_{i}, v_{j}])| = 4 or |V(C[υ_{i}, υ_{j}])| = 3. This means C[υ_{i}, υ_{j}] = υ_{i}υ^{+}_{i}υ^{−}_{i}υ_{j} or C[υ_{i}, υ_{j}] = υ_{i}υ^{−}_{i}υ_{j}. Let C′ = υ_{i}u_{1}υ_{j}υ^{−}_{j}υ^{+}_{j}C[υ^{+}_{j}, υ^{−}_{i}]υ^{−}_{i}υ^{+}_{i}υ_{i} or C′ = υ_{i}u_{1}υ_{j}υ^{−}_{j}υ^{+}_{j}C[υ^{+}_{j}, υ_{i}]. Then C′ is an o-cycle such that V(C) ⊂ V(C′) by Lemma 2.3 (d), contradicting Claim 1. □
Recall that υ^{–}_{i}υ^{+}_{i} ∈ E(G). Let H = G[{u_{1}, υ^{–}_{i}} ∪ V(C[υ_{i}, υ_{j}])] – {υ_{i}}. Since d_{G}(υ^{–}_{i}, u_{1}) = 2 and G is almost distance-hereditary, d_{H}(υ^{–}_{i}, u_{1}) ≤ 3. By Lemma 2.3 (c) and (d), we have υ^{–}_{i}υ_{j} ∉ E(G). By the choice of P, u_{1}υ ∉ E(G), where υ ∈ C[υ^{+}_{i}, υ^{–}_{j}]. It follows that d_{H}(υ^{–}_{i}, u_{1}) = 3 and d_{H}(υ^{–}_{i}, υ_{j}) = 2. By Lemma 2.3 (b), (c) and (d), υ^{–}_{i}υ^{–}_{j} ∉ E(G) and υ^{+}_{i}υ_{j} ∉ E(G). Thus there exists a vertex w ∈ C(υ^{+}_{i}, υ^{–}_{j}) such that υ^{–}_{i}w ∈ E(G) and wυ_{j} ∈ E(G). Note that w is well-defined.
wυ^{+}_{j} ∉ Ẽ(G).
Proof. Suppose wυ^{+}_{j} ∉ Ẽ(G). By Lemma 2.4 (a), we obtain υ^{–}_{i}w^{+} ∉ Ẽ(G). Since υ^{–}_{i}w ∈ E(G), we have υ_{j}w^{+} ∉ Ẽ(G) by Lemma 2.4 (b) and by symmetry. Note that υ_{j}υ^{–}_{i} ∉ Ẽ(G) by Lemma 2.3 (c). Thus {w, w^{+}, υ_{j}, υ^{–}_{i}} induces an o-light claw in G, a contradiction.
Next we will show that {υ_{j}, u_{1}, w, υ^{+}_{j}} induces an o-light claw and get a contradiction. Before proving this fact, the following claim is needed.
u_{1}w_{1} ∉ Ẽ(G).
Proof. First we will show that w^{–}υ^{–}_{i} ∉ Ẽ(G). Since υ^{–}_{j}υ^{+}_{j} ∈ Ẽ(G) and υ_{j} w ∈ E(G), we have w^{–}υ^{–}_{i} ∉ Ẽ(G) by Lemma 2.4 (b) and symmetry.
Next we will show that w^{−}υ_{j} ∈ Ẽ(G). Suppose not. Consider the subgraph induced by {w, w^{−}, υ_{j}, υ^{–}_{i}}. Note that υ_{j}υ^{–}_{i} ∉ Ẽ(G) by Lemma 2.3 (c) and w^{–}υ^{–}_{i} ∉ Ẽ(G) by the analysis above. Then {w, w^{−}, υ_{j}, υ^{–}_{i}} induces an o-light claw, a contradiction.
Now we will show that u_{1}w ∉ Ẽ(G), since otherwise, C′ = u_{1}wC[w, υ^{–}_{j}]υ^{–}_{j}υ^{+}_{j}C[υ^{+}_{j}, w^{–}]w^{–}υ_{j}u_{1} is an o-cycle such that V(C) ⊂ V(C′), contradicting Claim 1. □
By Claims 4, 5 and Lemma 2.3 (a), {υ_{j}, u_{1}, w, υ^{+}_{j}} induces an o-light claw, contradicting the fact G is claw-heavy. The proof of Theorem 1.3 is complete. □
Proof of Theorem 1.4.
Let G be a graph satisfying the condition of Theorem 1.4. By Lemma 2.1, there exists a heavy cycle in G. Now choose a longest heavy cycle C of G and assign an orientation to it. Suppose G is not Hamiltonian. Then
By the choice of C, all internal vertices of F are not heavy. By Lemma 2.3 (b), there is at most one heavy vertex in N^{+}_{C}(R) and at most one heavy vertex in N^{–}_{C}(R). Without loss of generality, assume that w^{–}_{i}, w^{+}_{i} are light. Hence w^{–}_{i}w^{+}_{i} ∈ E(G), otherwise {w_{i}, w^{–}_{i}, w^{+}_{i}, x_{r1}} induces a light claw, contradicting G is 1-heavy.
There exists a (u, C; 3)-fan F such that V(F) = {u, w_{i}, w_{j}, w_{k}}.
Proof. Now we choose the fan F in such a way that:
Q_{1} = uw_{i};
|V(C[w_{i}, w_{j}])| is as small as possible subject to (1);
|V(Q_{2})| is as small as possible subject to (1) and (2);
|V(C[w_{k}, w_{i}])| is as small as possible subject to (1), (2) and (3);
|V(Q_{3})| is as small as possible subject to (1), (2), (3) and (4).
Since G is 3-connected, for any neighbor of C in R, say u (with uw_{i} ∈ E(G), where w_{i} ∈ V(C)), there are three disjoint paths from u to C. Obviously, we can choose one such path as uw_{i}. Thus (1) is well-defined, and furthermore, the choice condition of F is well-defined.
V(Q_{2}) = {U, w_{j}}.
Proof. Suppose
V (Q_{3}) = {u, w_{k}}.
Proof. Suppose
If zw_{i} ∉ E(G), then set H = G[V (Q_{1}) ∪ V (Q_{3}) ∪ V(C[w_{k}, w_{i}])] − {w_{k}}. Since w^{–}_{i}w^{+}_{i} ∈ E(G), we obtain w^{+}_{k}w_{i} ∉ E(G) by Lemma 2.3 (c). This means d_{H} (w^{+}_{k}, w_{i}) ≥ 2. By the choice condition (4), we have
If zw_{i} ∈ E(G), then set H = G[V(C [w_{i}, w_{j}]) ∪ V(Q_{3} [u, z])] − {w_{i}}. Note that
By Claims 1.1 and 1.2, the proof of Claim 1 is complete. □
By Claim 1, there exists a (u, C; 3)-fan F such that V(F)\V(C) = {u}. Suppose that N_{C} (u) = {υ_{1}, υ_{2}, …, υ_{r}} (r ≥ 3) and υ_{1}, υ_{2}, …, υ_{r} are in the order of the orientation of C. In the following, all the subscripts of υ are taken modulo r, and υ_{0} = υ_{r}.
By Lemma 2.3 (b), there is at most one heavy vertex in N^{+}_{C}(u) and at most one heavy vertex in N^{–}_{C} (u). Since r ≥ 3, we know that there exists υ_{j} ∈ N_{C} (u), such that υ^{–}_{j}, υ^{+}_{j} are light, and hence υ^{–}_{j}υ^{+}_{j} ∈ E(G) by the fact G is 1-heavy. Without loss of generality, assume that υ^{–}_{1}υ^{+}_{1} ∈ E(G) and υ^{–}_{1}, υ^{+}_{1} are light. By Lemma 2.6 (b), there exists a vertex l_{1} ∈ C[υ^{+}_{1}, υ^{–}_{2}) such that υ^{–}_{2}l_{1} ∈ E(G) and l_{1}υ_{1} ∈ E(G), and there exists a vertex s_{1} ∈ C(υ^{+}_{0}, υ^{–}_{1}] such that υ^{+}_{0}s_{1} ∈ E(G) and s_{1}v_{1} ∈ E(G).
We divide the proof into two cases.
υ^{–}_{2}υ^{+}_{2} ∉ E(G) andυ^{–}_{0}μ^{+}_{0} ∉ E(G).
Both {υ, υ^{–}_{2}, υ^{+}_{2}, u} and {υ, υ^{–}_{0}, υ^{+}_{0}, u} induce claws. By Lemma 2.3 (b) and the fact G is 1-heavy, υ^{–}_{2} and υ^{+}_{0} are heavy or υ^{+}_{2} and υ^{–}_{0} are heavy.
υ^{–}_{1}l_{1} ∈ E(G) andl_{1}υ_{2} ∈ E(G).
Proof. Suppose υ^{–}_{1}l_{1} ∉ E(G). Note that uv^{–}_{1} ∉ Ẽ(G) by Lemma 2.3 (a). By Lemma 2.5 (a), l_{1} is light. Now {υ_{1}, l_{1}, u, υ^{–}_{1}} induces a light claw, a contradiction.
Suppose l_{1}v_{2} ∉ E(G). Let H = G[{υ^{–}_{1}, l_{1}, υ^{–}_{2}, υ_{2}, u}]. By Lemma 2.3, we get uv^{–}_{2} ∉ E(G), uv^{–}_{1} ∉ E(G) and υ^{–}_{1}υ^{–}_{2} ∉ E(G). Note that υ_{2}υ^{–}_{1} ∉ E(G) by Lemma 2.5 (a). Now G[{υ^{–}_{1}, l_{1}, υ^{–}_{2}, υ_{2}, u}] is an induced path of length 4 in G, contradicting Lemma 2.6 (a).
Now we consider the following two subcases.
υ^{–}_{2}, υ^{+}_{0}are heavy vertices.
By Lemma 2.3 (a), uυ^{+}_{2} ∉ Ẽ(G). By Lemma 2.5 (a), we have υ_{1}l^{–}_{1} ∈ E(G). Note that l′ ≔ l^{–}_{1} ∈ N(υ_{1}) and υ^{–}_{1}υ^{+}_{1} ∈ E(G). By Lemma 2.4 (b), l′^{+}υ^{+}_{2} = l_{1}υ^{+}_{2} ∉ Ẽ(G). By Lemma 2.5 (a) and Lemma 2.3 (b), l_{1} and υ^{+}_{2} are light. Now {υ_{2}, l_{1}, u, υ^{+}_{2}} induces a light claw, a contradiction.
υ^{+}_{2}, υ^{–}_{0}are heavy vertices.
Consider the subgraph induced by {υ^{+}_{0}, s_{1}, l_{1}, υ_{2}, υ}. It is easily to check that υ^{+}_{0}s_{1} ∈ E(G), l_{1}υ_{2} ∈ E(G) (by Claim 2) and υ_{2}u ∈ E(G). By Lemma 2.3 (a), υ^{+}_{0}u ∉ E(G). By Lemma 2.5 (a) and Lemma 2.4 (b), we know that υ_{1}l^{–}_{1} ∈ E(G) and υ^{+}_{0}l_{1} ∉ E(G). Now we obtain s_{1}l_{1} ∉ E(G) or υ^{+}_{0}υ_{2} ∈ E(G) or s_{1}υ_{2} ∈ E(G) (Otherwise, G[{υ^{+}_{0}, s_{1}, l_{1}, υ_{2}, u}] is an induced path of length 4, contradicting Lemma 2.6 (a)).
Suppose s_{1}l_{1} ∉ E(G). By Lemma 2.5 (a), l_{1}, s_{1} are light. Now {υ_{1}, l_{1}, s_{1}, u} induces a light claw, contradicting G is 1-heavy.
Suppose υ^{+}_{0}υ_{2} ∈ E(G). Consider the subgraph induced by {υ_{2}, υ^{–}_{2}, υ^{+}_{0}, u}. By Lemma 2.3 (a), we have uυ^{–}_{2} ∉ E(G) and uυ^{+}_{0} ∉ E(G). Since υ^{–}_{2}, υ^{+}_{0}, u are light and G is 1-heavy, υ^{+}_{0}υ^{–}_{2} ∈ E(G). By Lemma 2.5 (a) and Claim 2, υ_{1}l^{–}_{1} ∈ E(G) and l_{1}υ_{2} ∈ E(G). Now
Suppose s_{1}υ_{2} ∈ E(G). Consider the subgraph induced by {υ_{2}, υ^{–}_{2}, s_{1}, u}. By Lemma 2.5 (a) and Lemma 2.3 (b), s_{1} and υ^{–}_{2} are light. Since G is 1-heavy, s_{1}υ^{–}_{2} ∈ E(G). Now
υ^{–}_{2}υ^{+}_{2} ∈ E(G) orυ^{–}_{0}υ^{+}_{0} ∈ E(G).
Without loss of generality (by symmetry), assume that υ^{–}_{2}υ^{+}_{2} ∈ E(G).
υ^{–}_{0}υ^{+}_{0} ∉ E(G).
By Lemma 2.3 (a), uυ^{–}_{0} ∉ E(G) and uυ^{+}_{0} ∉ E(G). Now {υ_{0}, υ^{–}_{0}, υ^{+}_{0}, u} induces a claw. Since G is 1-heavy and u is light, υ^{–}_{0} is heavy or υ^{+}_{0} is heavy.
Suppose υ^{–}_{0} is heavy. By Lemma 2.3 (b), (c) and Lemma 2.4 (b), υ^{–}_{2}, υ_{1} and l^{–}_{1} are light. By Lemma 2.6 (d), {l_{1}, l^{–}_{1}, υ_{1}, υ^{–}_{2}} induces a light claw, a contradiction.
Suppose υ^{+}_{0} is heavy. By Lemma 2.3 (b), (c) and Lemma 2.4 (b), we can see υ^{+}_{2}, υ_{1} and l^{+}_{1} are light. By Lemma 2.6 (d), {l_{1}, l^{+}_{1}, υ_{1}, υ^{+}_{2}} induces a light claw, a contradiction. (Note that υ^{–}_{1}υ^{+}_{1} ∈ E(G) and υ^{–}_{2}υ^{+}_{2} ∈ E(G). By Lemma 6 (c), l_{1}υ^{+}_{2} ∈ E(G).
υ^{–}_{0}υ^{+}_{0} ∈ E(G).
By Lemma 6 (b), there exists a vertex s_{2} ∈ C(υ^{+}_{1}, υ^{–}_{2}] such that υ^{+}_{1}s_{2} ∈ E(G) and s_{2}υ_{2} ∈ E(G).
(i) υ_{1}is heavy, (ii) l^{–}_{0}, l^{+}_{0}, s^{–}_{2}, s^{+}_{2}are light.
Proof. Recall that the definition of l_{0} occurred in the condition of Lemma 5 before. Let l_{0} ∈ C[υ^{+}_{0}, υ^{–}_{1}) such that υ^{–}_{0}l_{0} ∈ E(G) and l_{0}υ_{0} ∈ E(G).
(i) By Lemma 2.6 (d), each of {l_{0}, l^{–}_{0}, υ_{0}, υ^{–}_{1}} and {l_{1}, l^{–}_{1}, υ_{1}, υ^{–}_{2}} induces a claw. Since G is 1-heavy, at least one vertex of {l^{–}_{1}, υ_{1}, υ^{–}_{2}} is heavy.
Suppose υ^{–}_{2} is heavy. By Lemma 2.3 (b), (c) and Lemma 2.4 (b), υ^{–}_{1}, υ_{0} and l^{–}_{0} are light. Now {l_{0}, l^{–}_{0}, υ_{0}, υ^{–}_{1}} induces a light claw, contradicting G is 1-heavy.
Suppose l^{–}_{1} is heavy. By Lemma 2.6 (c), υ^{+}_{2}l_{1} ∈ E(G). By Lemma 2.4 (a) and (b), υ^{–}_{1}l^{–}_{1} ∉ Ẽ(G) and υ_{0}l^{–}_{1} ∉ Ẽ(G). This implies that υ_{0}, υ^{–}_{1} are light. At the same time, we can prove that l_{0}^{–} is light. (Otherwise,
Note that {l_{1}, l^{–}_{1}, υ_{1}, υ^{–}_{2}} induces a claw and υ^{–}_{2}, l^{–}_{1} are light. Since G is 1-heavy, υ_{1} is heavy.
(ii) Note that υ_{1} is heavy. If l^{+}_{0} is heavy, then C′ = υ_{0}uυ_{1}l^{+}_{0}C[l^{+}_{0}, υ^{–}_{1}]υ^{–}_{1}υ^{+}_{1}C[υ^{+}_{1}, υ^{–}_{0}]υ^{–}_{0}υ^{+}_{0}C[υ^{+}_{0}, l_{0}]l_{0}υ_{0} is an o-cycle such that V(C) ⊂ V(C′), a contradiction. If l^{–}_{0} is heavy, then C′ = υ_{0}l_{0}C[l_{0}, υ^{–}_{1}]υ^{–}_{1}υ^{+}_{1}C[υ^{+}_{1}, υ^{–}_{0}]υ^{–}_{0}υ^{+}_{0}C[υ^{+}_{0}, l^{–}_{0}]l^{–}_{0}υ_{1}uυ_{0} is an o-cycle such that V(C) ⊂ V(C′), a contradiction. Similarly, by symmetry, we can prove that s^{–}_{2}, s^{+}_{2} are light. □
υ^{–}_{1}l^{+}_{0} ∈ E(G) andυ^{–}_{1}s^{+}_{2} ∈ E(G).
Proof. Suppose υ^{–}_{1}l^{+}_{0} ∉ E(G). By Lemma 2.4 (b) and (c), υ^{–}_{1}l^{–}_{0} ∉ Ẽ(G) and l^{–}_{0}l^{+}_{0} ∉ Ẽ(G). By Claim 3, l^{+}_{0}, l^{–}_{0} are light. Now {l_{0}, l^{–}_{0}, l^{+}_{0}, υ^{–}_{1}} induces a light claw, a contradiction.
By Lemma 2.6 (c) and by symmetry, we obtain υ^{–}_{1}s_{2} ∈ E(G). Suppose υ^{–}_{1}s^{+}_{2} ∉ E(G). By Lemma 2.4 (b) and (c), υ^{–}_{1}s^{–}_{2} ∉ Ẽ(G) and s^{–}_{2}s^{+}_{2} ∉ Ẽ(G). By Claim 3, s^{+}_{2}, s^{–}_{2} are light. Now {s_{2}, s^{–}_{2}, s^{+}_{2}, υ^{–}_{1}} induces a light claw, a contradiction. □
By Claim 3, Lemma 2.5 (b) and Claim 4, l^{+}_{0}, s^{+}_{2} are light,
The proof of Theorem 1.4 is complete. □
The authors are particularly grateful to the anonymous referee for his/her extensive comments that considerably improved the paper.
The first author is supported by NSFC (No. 11271300) and the Scientific Research Program of Shaanxi Provincial Education Department (No. 2013JK0580).
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