Hamilton cycles in almost distance-hereditary graphs

1 Introduction

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 NP-complete, which is shown by Karp [2]. However, if we only consider some restricted graph classes, then the situation is completely changed. A graph G is called distance-hereditary if each connected induced subgraph H has the property that d_H(x, y) = d_G(x, y) for any pair of vertices x, y in H. This concept was introduced by Howorka [3] and a complete characterization of distance-hereditary graphs could be found in [3]. In 2002, Hsieh, Ho, Hsu and Ko [4] obtained an O(|V| + |E|)-time algorithm to solve the Hamiltonian problem on distance-hereditary graphs. Some other optimization problems can also be solved in linear time for distance-hereditary graphs although they are proved to be NP-hard for general graphs. For references in this direction, we refer to [5, 6].

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.

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] 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.

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].

We postpone the proofs of Theorem 1.3 and 1.4 to Section 3.

Fig. 1.

A graph which shows the result in Theorem 1.3 strengthens that in Theorem 1.2.

2 Preliminaries

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 = υ₁υ₂ … υ_k υ₁ is called an Ore-cycle or briefly, o-cycle of G, if we have υ_iυ_i+1 ∈ Ẽ(G) for every i ∈ {1, 2, · , k}, where υ₁ = 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 P₁, P₂, … , 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: P₁, P₂, … , P_k) to denote the fan. The vertices in V(F)\{w₁, w₂, … , 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 H← the same graph as H but with the reverse orientation. For a vertex υ ∈ V(H), we use υ_H⁺ to denote the successor of υ on H, and υ_H⁻ to denote its predecessor. If S ⊆ V(H), then define S⁺_H = {υ⁺_H : υ ∈ S} and S⁻_H = {υ⁻_H : υ ∈ S}. If there is no danger of ambiguity, we denote υ⁺_H, υ⁻_H, S⁺_H and S⁻_H by υ⁺, υ⁻, S⁺ and S⁻, respectively. For two vertices u, υ ∈ V(H), we denote by H[u, υ] the segment of H from u to υ, and denote H(u, υ), H [u, υ) and H(u, υ] by the paths H[u, υ] − {u, υ}, H[u, υ] − {υ} and H[u, υ] − {u}, respectively.

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.

Proof. (a) Suppose that uυ⁻_i ∈ Ẽ(G). Then C′ = uυ_iC[υ_i, υ⁻_i]υ⁻_iu is an o-cycle of length longer than C, a contradiction. The other assertion can be proved similarly.

(b) Suppose that υ⁻_iυ⁻_j ∈ Ẽ(G). Then C′=uυjC[υj,υi−]υi−υj−C←[υj−,υi]υiu is an o-cycle of length longer than C, a contradiction. The other assertion can be proved similarly.

(c) Suppose that υ_iυ⁻_j ∈ Ẽ(G). Then C′ = υ_iuυ_jC[υ_j, υ⁻_i]υ⁻_iυ⁺_iC[υ⁺_i, υ⁻_i]υ⁻_iυ_i is an o-cycle of length longer than C, a contradiction. The other assertion can be proved similarly.

(d) If υ⁻_iυ⁺_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⁻_iυ⁺_i ∈ Ẽ(G). If υ⁻_iυ⁺_i ∈ E(G), then obviously υ⁻_iυ⁺_i ∈ Ẽ(G). The other assertion can be proved similarly.

(e) Suppose that υ⁻_iυ⁺_i ∉ E(G) and υ⁻_jv⁺_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⁻_iυ⁻_j ∈ Ẽ(G) or υ⁺_iυ⁺_j ∈ Ẽ(G), a contradiction to (b).

Proof. Let P be a (υ_i, υ_j)-path with all internal vertices in R.

(a) Suppose l⁻υ⁺_j ∈ Ẽ(G). Then C' = P←C[υi, l−]l−υj+C[υj+, υi−]υi−lC[l, υj] is an o-cycle such that V(C) ⊂ V(C′). By Lemma 2.2, there is a longer cycle C″ containing all vertices in C, that is, a longer cycle (a longer heavy cycle) in G, contradicting the choice of C. Suppose l⁺υ⁺_j ∈ Ẽ(G). Then C' = PC←[υj, l+]l+υj+C[υj+, υi−]υi−lC←[l, υi] is an o-cycle such that V(C) ⊂ V(C′), a contradiction.

(b) Suppose l⁻υ⁻_j ∈ Ẽ(G). Then C' = PC[υj, υi−]υi−υi+C[υi+,l−]l−υj−C←[υj−, l]lυi is an o-cycle such that V(C) ⊂ V(C′), a contradiction. Suppose l⁺υ⁺_j ∈ Ẽ(G). Then C' = PC←[υj, l+]l+υj+C[υj+, υi−]υi−υi+C[υi+,l]lυi is an o-cycle such that V(C) ⊂ V(C′), a contradiction.

(c) Suppose l⁻l⁺ ∈ Ẽ(G). Then C′ = PC[υ_j, υ⁻_i]υ⁻_iυ⁺_iC[υ⁺_i, l⁻]l⁻l⁺C[l⁺, υ⁻_j]υ⁻_jlυ_i is an o-cycle such that V(C) ⊂ V(C′), a contradiction. □

Proof. By Lemma 2.3 (a), uυ⁻₁ ∉ E(G) and uυ⁺₁ ∉ E(G). If υ⁻₁υ⁺₁ ∉ E(G), then since u, υ⁻₁, υ⁺₁ are light, {υ₁, u, υ⁻₁, υ⁺₁} induces a light claw, a contradiction. Thus υ⁻₁υ⁺₁ ∈ E(G).

(a) Since υ⁻₂υ⁺₂ ∉ E(G) and υ⁻₀υ⁺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⁻₁υ₂ ∈ E(G) and υ⁻₂, υ⁺₀ are heavy. Let C′ = υ⁻₁υ₂C[υ₂, υ₀]υ₂uυ₁C[υ₁, υ⁻₂]υ⁻₂υ⁺₀C[υ⁺₀, υ⁻₁]. Then C′ is an o-cycle such that V(C) ⊂ V(C′), a contradiction.

Suppose υ⁻₁υ₂ ∈ E(G) and υ⁺₂, υ⁻₀ are heavy. Now {υ₂, υ⁻₂, u, υ⁻₁} induces a light claw, a contradiction.

(ii) Suppose υ₁l⁻₁ ∉ E(G). Note that υ₁l⁺₁ ∉ Ẽ(G) and l⁻₁l⁺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), υ₁, l⁺₁, l⁻₁ are light. Now {l₁, l⁺₁, υ₁, l⁻₁} induces a light claw, a contradiction. Similarly, we can prove that υ₁s⁺₁ ∈ E(G).

(iii) By Lemma 2.3 (c) and Lemma 2.4 (b), υ₁, l⁺₁, l⁻₁, s⁻₁, s⁺₁ are light. Since υ₁l⁻₁ ∈ E(G), we obtain υ⁺₂l₁ ∉ Ẽ(G) and υ⁺₀l₁ ∉ Ẽ(G) by Lemma 2.4 (b). Note that either υ⁺₀ or υ⁺₂ is a heavy vertex. This implies l₁ is a light vertex. The other assertion that s₁ is light can be proved similarly.

(b) Since υ₀l₀ ∈ E(G) and υ₂s₂ ∈ E(G), υ⁺₁l⁺₀ ∉ Ẽ(G) and υ⁺₁s⁺₂ ∈ Ẽ(G) by Lemma 2.4 (b). Furthermore, we can prove that l⁺₀s⁺₂ ∉ E(G). (Otherwise, C'= υ0uυ2s2C←[s2,l0+]l0+s2+C[s2+,υ2−]υ2−υ2+C[υ2+,υ0−]υ0−υ0+C[υ0+,l0]l0υ0 is an o-cycle such that V(C) ⊂ V(C′), a contradiction.)

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⁻_iυ⁺_i ∈ Ẽ(G), by Lemma 2.3 (c), we have υ_iυ⁻_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+1l_i ∈ E(G) and l_i υ_i ∈ E(G). The other assertion can be proved similarly.

(c) Suppose υ⁺_i+1l_i ∉ E(G). Let H = G[{υ⁺_i+1, υ⁻_i+1, l_i, υ_i, u}]. By Lemma 2.3 (c), υ_iυ⁻_i+1 ∉ E(G) and υ_iυ⁺_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_iυ⁻_i+1 ∉ Ẽ(G) and l⁻_iυ⁻_i+1 ∉ Ẽ(G). By Lemma 2.6 (c) and Lemma 2.4 (b), we have υ_il⁻_i ∉ E(G). So {l_i, l⁻_i, υ_i, υ⁻_i+1} induces a claw. The other assertion can be proved similarly. □

3 Proofs of Theorems 1.3 and 1.4

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(G)\V(C)≠0. Let R be a component of G − C, and A = {υ₁, υ₂, … , υ_k} be the set of neighbors of R on C. Since G is 2-connected, there exists a (υ_i, υ_j)-path P = υ_iu₁ … u_r υ_j with all internal vertices in R, and υ_i, υ_j ∈ A. Choose P such that:

1. |V(C(υ_i, υ_j))| is as small as possible;

2. |V(P)| is as small as possible subject to (1).

Proof. First we will show that w^–υ^–_i ∉ Ẽ(G). Since υ^–_jυ⁺_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_jυ^–_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 u₁w ∉ Ẽ(G), since otherwise, C′ = u₁wC[w, υ^–_j]υ^–_jυ⁺_jC[υ⁺_j, w^–]w^–υ_ju₁ is an o-cycle such that V(C) ⊂ V(C′), contradicting Claim 1. □

By Claims 4, 5 and Lemma 2.3 (a), {υ_j, u₁, 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 V(G)\V(C)≠0. Let R be a component of G − C and A = {w₁, w₂, … , w_k} be the set of neighbors of R on C. Since G is 3-connected, for any vertex u of R, there exists a (u, C; 3)-fan F such that F = (u; Q₁, Q₂, Q₃), where Q₁ = ux₁ … x_r1w_i, Q₂ = uy₁ … y_r2w_j and Q₃ = uz₁ … z_r3w_k, and w_i, w_j, w_k are in the order of the orientation of C.

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^–_iw⁺_i ∈ E(G), otherwise {w_i, w^–_i, w⁺_i, x_r1} induces a light claw, contradicting G is 1-heavy.

Proof. Suppose V(Q2)\{u,wj}≠0. Without loss of generality, set y = y_r2. Let H = G[V(Q₁) ∪ V(Q₂) ∪ V(C[w_i, w_j])] − {w_j}. Note that w^–_iw⁺_i ∈ E(G). By Lemma 2.3 (c), it is easy to see that w_i w^–_j ∉ E(G), so d_H(w^–_j, w_i) ≥ 2. In the meantime, the choice condition (2) implies that N(V(Q2)\{wj})∩V(C(wj,wj))=0. This means that d_H (w^–_j, y) = d_H (w^–_j, w_i) + d_H (w_i, y) ≥ 2 + d_H (w_i, y). Since G is almost distance-hereditary and d_G (w^–_j, y) = 2, we have d_H (w^–_j, y) = 3 and yw_i ∈ E(G). Let F′ = (y; Q′₁, Q′₂, Q′₃) such that Q′₁ = yw_i, Q′₂ = yw_j and Q′₃ = Q₂[y, u]Q₃[u, w_k]. Then F′ is a (y, C;3)-fan satisfying (1), (2) and |V(Q′₂)| = 2, contradicting the choice condition (3), a contradiction. □

Proof. Suppose V(Q3)\{u,wk}≠0. Without loss of generality, set z = z_r3.

If zw_i ∉ E(G), then set H = G[V (Q₁) ∪ V (Q₃) ∪ V(C[w_k, w_i])] − {w_k}. Since w^–_iw⁺_i ∈ E(G), we obtain w⁺_kw_i ∉ E(G) by Lemma 2.3 (c). This means d_H (w⁺_k, w_i) ≥ 2. By the choice condition (4), we have N(V(Q3)\{wk})∩V(C(wk,wi))=0, and hence d_H (w⁺_k, z) = d_H (w⁺_k, w_i) + d_H (w_i, z). Since zw_i ∉ E(G), d_H (w_i, z) ≥ 2 and we get d_H (w⁺_k, z) ≥ 4. It yields a contradiction to the fact G is almost distance-hereditary and d_G (w⁺_k, z) = 2.

If zw_i ∈ E(G), then set H = G[V(C [w_i, w_j]) ∪ V(Q₃ [u, z])] − {w_i}. Note that NC(z)∩V(C(wi,wj])=0 (by the choice conditions (2), (5)) and NC(V(Q3)\{z,wk})∩V(C(wi,wj))=0 (by the choice condition (2)). Since d_G(w⁺_i, z) = 2 and G is almost distance-hereditary, d_H(w⁺_i, z) ≤ 3. But if w⁺_i, w_j ∉ E(G), then the distance from z to w⁺_i in H is at least 4, where in such a shortest path, the path Q₃ [z, u] contributes at least 1, the path Q₂ [u, w_j] contributes 1, a contradiction. Thus we have w⁺_iw_j ∈ E(G), and hence w^–_jw⁺_j ∉ Ẽ(G) by Lemma 2.3 (c). Consider the subgraph induced by {w_j, w⁺_i, w⁺_j, u}. Since G is 1-heavy and w⁺_i, u are light, w⁺_j is heavy. Now let H = G[{w^–_i} ∪ V(C[w_i, w_j]) ∪ V(Q₃[u, z])] – {w_i}. Similarly, since d_G(w^–_i, z) = 2, we have d_H(w^–_i, z) = 3, and w^–_i, w_j ∈ E(G). Consider the subgraph induced by {w_j, w^–_i, w^–_j, u}. Similarly, we can see w^–_j is heavy, and hence w^–_jw⁺_j ∈ Ẽ(G), a contradiction. Thus V(Q₃) = {u, w_k}. □

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) = {υ₁, υ₂, …, υ_r} (r ≥ 3) and υ₁, υ₂, …, υ_r are in the order of the orientation of C. In the following, all the subscripts of υ are taken modulo r, and υ₀ = 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^–_jυ⁺_j ∈ E(G) by the fact G is 1-heavy. Without loss of generality, assume that υ^–₁υ⁺₁ ∈ E(G) and υ^–₁, υ⁺₁ are light. By Lemma 2.6 (b), there exists a vertex l₁ ∈ C[υ⁺₁, υ^–₂) such that υ^–₂l₁ ∈ E(G) and l₁υ₁ ∈ E(G), and there exists a vertex s₁ ∈ C(υ⁺₀, υ^–₁] such that υ⁺₀s₁ ∈ E(G) and s₁v₁ ∈ E(G).

We divide the proof into two cases.

Proof. Suppose υ^–₁l⁺₀ ∉ E(G). By Lemma 2.4 (b) and (c), υ^–₁l^–₀ ∉ Ẽ(G) and l^–₀l⁺₀ ∉ Ẽ(G). By Claim 3, l⁺₀, l^–₀ are light. Now {l₀, l^–₀, l⁺₀, υ^–₁} induces a light claw, a contradiction.

By Lemma 2.6 (c) and by symmetry, we obtain υ^–₁s₂ ∈ E(G). Suppose υ^–₁s⁺₂ ∉ E(G). By Lemma 2.4 (b) and (c), υ^–₁s^–₂ ∉ Ẽ(G) and s^–₂s⁺₂ ∉ Ẽ(G). By Claim 3, s⁺₂, s^–₂ are light. Now {s₂, 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, {υ1+l0+,υ1+s2+,l0+s2+}∩E(G)=0 and {υ^–₁l⁺₀, υ^–₁s⁺₂} ⊂ E(G). It is proved that {υ^–₁, υ⁺₁, l⁺₀, s⁺₀} induces a light claw, a contradiction.

The proof of Theorem 1.4 is complete. □