Some sufficient conditions on hamilton graphs with toughness

Let G be a graph, and the number of components of G is denoted by c(G). Let t be a positive real number. A connected graph G is t-tough if tc(G − S) ≤ |S| for every vertex cut S of V(G). The toughness of G is the largest value of t for which G is t-tough, denoted by τ(G). We call a graph G Hamiltonian if it has a cycle that contains all vertices of G. Chvátal and other scholars investigate the relationship between toughness conditions and the existence of cyclic structures. In this paper, we establish some sufficient conditions that a graph with toughness is Hamiltonian based on the number of edges, spectral radius, and signless Laplacian spectral radius of the graph.

MR subject classifications: 05C50, 15A18.

1. Introduction

Let G = [V(G), E(G)] be a finite simple undirected graph with vertex set V(G) = {v₁, v₂, …, v_n} and edge set E(G). Write by m = |E(G)| the number of edges and n = |V(G)| the number of vertices of the graph G, respectively. The set of neighbors of a vertex v in graph G is denoted by N_G(v). Let v_i ∈ V(G), we denote by d_i = d_vi = d_G(v_i) = |N_G(v_i)| the degree of v_i. Denote by δ(G) [Δ(G)] or simply δ (Δ) the minimum (maximum) degree of G. Let (d₁, d₂, …, d_n) be a nondecreasing degree sequence of G, that is, d₁ ≤ d₂ ≤ ⋯ ≤ d_n. For convenience, we use $(0^{x_0},1^{x_1},\cdots ,k^{x_k},\cdots ,{\text{Δ}}^{x_{\text{Δ}}})$ to denote the degree sequence of G, where x_k is the number of vertices of degree k in the graph G. We denote a bipartite graph with bipartition (X, Y) by using G[X, Y]. We denote the cycle and the complete graph on n vertices by using C_n and K_n, respectively. We use K_{m, n} to denote a complete bipartite graph with two parts having m, n vertices, respectively. Let G and H be two disjoint graphs. We denote by G + H the disjoint union of G and H, which is a graph with vertex set V(G) ∪ V(H) and edge set E(G) ∪ E(H). If G₁ = G₂ = … = G_k, we denote G₁ + G₂ + ⋯ + G_k by kG₁. We denote by G ∨ H the join of G and H, which is a graph obtained from the disjoint union of G and H by adding edges joining every vertex of G to every vertex of H. Let K_{n − 1} + v denote the complete graph K_{n − 1} together with an isolated vertex v. Other undefined symbols reference can be seen in Bondy and Murty (1982) and Bauer et al. (2006).

The adjacency matrix of G is A(G) = (a_ij), where a_ij = 1 if v_i and v_j are adjacent in G and a_ij = 0 otherwise. Let D(G) be the degree diagonal matrix of G, i.e., D(G) = diag{d_G(v₁), d_G(v₂), …, d_G(v_n)}. The matrix Q(G) = D(G) + A(G) is called the signless Laplacian matrix of G. The largest eigenvalue of A(G), denoted by μ(G), is called to be the spectral radius of G. The largest eigenvalue of Q(G), denoted by q(G), is called to be the signless Laplacian spectral radius of G.

A cycle (path) containing every vertex of a graph is called a Hamilton cycle (path) of the graph. Graph G is called a Hamilton graph if it has a Hamilton cycle, and then we also call G Hamiltonian. The number of components of G is denoted by c(G). Let t be a positive real number. A connected graph G is t-tough if tc(G − S) ≤ |S| for every vertex cut S of V(G). The toughness of G is the largest value of t for which G is t-tough, denoted by τ(G). If G is a complete graph, take τ(K_n) = ∞ for all n ≥ 1. If G is not a complete graph, $\tau (G)=min\{\frac{\left|S\right|}{c(G-S)}:S\subseteq V(G),c(G-S)\ge 2\}$, where the minimum is taken over all cut sets of vertices in G. Obviously, a t-tough graph is s-tough for all s < t.

On the one hand, more than 40 years ago, Chvátal (1973) introduced the concept of toughness. From then on a lot of research has been obtained, mainly relating to the relationship between toughness conditions and the existence of cyclic structures. Historically, most of the research was based on some conjectures in Chvátal (1973). D. Bauer etc. (Bauer et al., 1991, 1995a,b, 1999), surveyed results on toughness and its relationship to cycle structure. If we want to know more about the Hamiltonian problems related to toughness, we can refer to Bauer et al. (2006) and Huang et al. (2022). On the other hand, the problem of determining whether a graph is Hamiltonian is an NP-complete problem. In recent years, the study of Hamiltonian problem using spectrum graph theory has received extensive attention, and some meaningful results are obtained, such as Fiedler and Nikiforov (2010), Zhou (2010), Lu et al. (2012), Yu and Fan (2013), Liu et al. (2015), Li and Ning (2016), Feng et al. (2017), Zhou et al. (2018), and Yu et al. (2019). We would naturally think of what a t-tough graph is Hamiltonian when adding to other conditions. Inspired by the above results, in this paper, we establish some sufficient conditions that a graph with toughness is Hamiltonian based on the number of edges, spectral radius, and signless Laplacian spectral radius of the graph.

2. Preliminary

At the beginning of this section, we first give some definitions. Let G be a graph on n vertices. A vector X ∈ Rⁿ is called to be defined on the vertex set V(G) of the graph G, if there is a one-to-one mapping φ from vertex set V(G) of the graph to the components of the vector X; simply written X_u = φ(u).

When μ is an eigenvalue of the adjacency matrix A(G) corresponding to the eigenvector X if and only if X ≠ 0,

$\begin{array}{ll}\mu X_v={\displaystyle \sum _{w\in N_G(v)}{X}_w,\text{for each vertex }v\in V(G).}& (2.1)\end{array}$

The Equation (2.1) is called the characteristic equation of G.

When q is an eigenvalue of signless Laplacian matrix Q(G) corresponding to the eigenvector X if and only if X ≠ 0,

$\begin{array}{ll}[q-d_G(v)]X_v={\displaystyle \sum _{w\in N_G(v)}{X}_w,\text{for each vertex }v\in V(G).}& (2.2)\end{array}$

The Equation (2.2) is called the signless Laplacian characteristic equation of G.

Lemma 2.1. Hoàng (1995) let t ∈ {1, 2, 3} and G be a t-tough graph with a non-decreasing degree sequence d₁ ≤ d₂ ≤ ⋯ ≤ d_n. If for all integers k with $t\le k<\frac{n}{2}$, d_k ≤ k implies d_n−k+t ≥ n − k, then G has a Hamilton cycle.

Lemma 2.2. Yuan (1988) let G be a connected graph with n vertices and m edges. Then

$\mu (G)\le \sqrt{2m-n+1,}$

and the equality holds if and only if G = K_n or G = K_1,n−1.

Lemma 2.3. Yu and Fan (2013) let G be a graph with n vertices and m edges. Then

$q(G)\le \frac{2m}{n-1}+n-2.$

If G is connected, the equality holds if and only if G = K_1,n−1 or G = K_n. Otherwise, the equality holds if and only if G = K_n−1+v.

Lemma 2.4. Hoàng (1995) every Hamiltonian graph is 1-tough.

Lemma 2.5. Let the graph G be not a complete graph with minimum degree δ(G), and G is t-tough, then δ(G) ≥ 2t.

Proof Let δ(G) = d_v, S = N_G(v), then |S| = |N_G(v)| = δ(G). We can get c(G − S) ≥ 2. Because G is not a complete graph, by

$\tau (G)=min\{\frac{\left|S\right|}{c(G-S)}:S\subseteq V(G),c(G-S)\ge 2\},$

then

$t\le \tau (G)\le \frac{\left|S\right|}{c(G-S)}\le \frac{\delta (G)}{2},$

thus, we can get δ(G) ≥ 2t.

The proof is completed.■

Lemma 2.6. Jung (1978) let G be a graph without a Hamiltonian cycle and at least 11 vertices. Then

(i) there exist two non-adjacent vertices x, y such that d(x) + d(y) ≤ |V(G)| − 5 or

(ii) there exist for some t ≥ 1 vertices x₁, x₂, …, x_t such that G − x₁ − ⋯ − x_t has at least t + 1 components.

Corollary 2.7. Let t ∈ {1, 2, 3}, and G be a t-tough graph without a Hamiltonian cycle with at least 11 vertices. Then there exist two non-adjacent vertices x, y such that d(x) + d(y) ≤ |V(G)| − 5.

Proof Because G has no Hamiltonian cycle, G is not a complete graph. If there exist for some s ≥ 1 vertices x₁, x₂, …, x_s such that G − x₁ − ⋯ −x_s has at least s + 1 components, by

$\tau (G)=min\{\frac{\left|S\right|}{c(G-S)}:S\subseteq V(G),c(G-S)\ge 2\},$

then

$t\le \tau (G)\le \frac{s}{s+1}<1,$

a contradiction.

The proof is completed.■

The closure of a graph G, denoting by ${{C}}_n(G)$, is the graph obtained from G by recursively joining pairs of nonadjacent vertices whose degree sum is at least n until no such pair remains, refer to Bondy and Chvátal (1976).

Lemma 2.8. Bondy and Chvátal (1976) a graph G is Hamiltonian if and only if ${{C}}_n(G)$ is Hamiltonian.

3. Main results

Theorem 3.1. Let G be a t-tough (t ∈ {1, 2, 3}) and simple connected graph with n(≥ 8t) vertices and m edges. If

$\begin{array}{ll}m\ge \left(\begin{array}{c}n-2t\\ 2\end{array}\right)+3t^2,& (3.1)\end{array}$

then

(i) G is Hamiltonian when t ∈ {1, 2} and n ≥ 8t.

(ii) G is Hamiltonian when t = 3 and n > 9t.

Proof Suppose that G is not a Hamilton graph. By Lemma 2.1, there exists a positive integer k for $t\le k<\frac{n}{2}$ and d_k ≤ k, such that d_{n − k + t} ≤ n − k − 1. Then we have

$\begin{array}{l}\begin{array}{ccc}\hfill 2m& ={\displaystyle \sum _{i=1}^k}{d}_i+{\displaystyle \sum _{i=k+1}^{n-k+t}}{d}_i+{\displaystyle \sum _{i=n-k+t+1}^n}{d}_i\hfill & \hfill \\ \hfill & \le k^2+(n-2k+t)(n-k-1)+(k-t)(n-1)\hfill \\ \hfill & =n^2-n+3k^2+(1-2n-t)k\hfill \\ \hfill & =2\left(\begin{array}{c}n-2t\\ 2\\ \end{array}\right)+6t^2-(k-2t)(2n-3k-5t-1),\hfill \end{array}\end{array}$

thus

$\begin{array}{ll}m\le \left(\begin{array}{c}n-2t\\ 2\end{array}\right)+3t^2-\frac{(k-2t)(2n-3k-5t-1)}{2}.& (3.2)\end{array}$

Since $\left(\begin{array}{c}n-2t\\ 2\\ \end{array}\right)+3t^2\le m\le \left(\begin{array}{c}n-2t\\ 2\\ \end{array}\right)+3t^2-\frac{(k-2t)(2n-3k-5t-1)}{2}$, thus (k − 2t)(2n − 3k − 5t − 1) ≤ 0. Next, we discuss three cases.

Case 1 t = 1.

In this case, n ≥ 8t = 8, (k − 2)(2n − 3k − 6) ≤ 0. By Lemma 2.5, δ (G)≥ 2t = 2. Since δ(G) ≤ d_k ≤ k, then k ≥ 2.

Case 1.1 (k − 2)(2n − 3k − 6) = 0, i.e., k = 2; or k ≠ 2 and 2n − 3k − 6 = 0.

Case 1.1.1 k = 2.

In this case, G is a graph with d₂ ≤ 2, d_{n − 1} ≤ n−3, d_n ≤ n − 1, and we have $(n-2)(n-3)+6\le {\displaystyle \sum _{i=1}^n{d}_i\le (n-2)(n-3)+6}$ by (3.1) and (3.2). Thus, all inequalities of (3.2) become equality. In this time, G must be with degree sequence [2², (n − 3)^{n − 3}, n − 1].

If two 2-degree vertices of G are non-adjacent, G must be the graph K₁ ∨ (K_{n − 3} − uv + wu + zv), where u, v ∈ V(K_{n − 3}), w, z ∉ V(K_{n − 3}), is Hamiltonian, a contraction.

If two 2-degree vertices of G are adjacent, G must be the graph K₁ ∨ (K₂ + K_{n − 3}), but $\tau \left[K_1\vee (K_2+K_{n-3})\right]\le \frac{1}{2},$ a contraction.

Case 1.1.2 k ≠ 2 and 2n − 3k − 6 = 0.

In this case, we can get n ≤ 11 because $k<\frac{n}{2}$, and hence n = 9, k = 4. Then d₄ ≤ 4, d₆ ≤ 4, d₉ ≤ 8, and we have $48\le {\displaystyle \sum _{i=1}^9{d}_i\le 48}$ by (3.1) and (3.2). Thus, all inequalities of (3.2) become equality. G must be with degree sequence (4⁶, 8³), and G = K₃ ∨ (3K₂) is Hamiltonian, a contradiction.

Case 1.2 (k − 2)(2n − 3k − 6) < 0.

In this case, k ≥ 3 and 2n − 3k − 6 < 0. Since $k<\frac{n}{2}$, we have n ≥ 2k + 1. From these results, we have 4k + 2 ≤ 2n ≤ 3k + 5, that is k ≤ 3. Thus, we have k = 3, n = 7. A contradiction with known condition n ≥ 8.

Case 2 t = 2.

In this case, (k − 4)(2n − 3k − 11) ≤ 0. By Lemma 2.5, δ(G)≥ 2t = 4. Since δ(G) ≤ d_k ≤ k, then k ≥ 4.

Case 2.1 (k − 4)(2n − 3k − 11) = 0, i.e., k = 4; or k ≠ 4 and 2n − 3k − 11 = 0.

Case 2.1.1 k = 4.

In this case, G is a graph with d₄ ≤ 4, d_{n − 2} ≤ n − 5, d_n ≤ n − 1, and we have $(n-4)(n-5)+24\le {\displaystyle \sum _{i=1}^n{d}_i\le (n-4)(n-5)+24}$ by (3.1) and (3.2). Thus, all inequalities of (3.2) become equality. During this time, G is the graph with degree sequence (4⁴, (n − 5)^{n − 6}, (n − 1)²), and n ≥ 8t = 16 for t = 2. Let S is the set containing four 4 − degree vertices of G, and there exist two non-adjacent vertices in S by Corollary 2.7.

By Lemma 2.8, ${{C}}_n(G)$ is also not Hamiltonian. According to the definition of ${{C}}_n(G)$, all points except the vertices of S form a complete graph K_{n − 4} in ${{C}}_n(G)$. Let us discuss graph ${{C}}_n(G)$. If one vertex of the K_{n − 4} is adjacent to one vertex of S, it must be adjacent to all vertices of S. Moreover, there are at least 2 vertices of the K_{n − 4} adjacent to all vertices of S because there exist two non-adjacent vertices in S by Corollary 2.7.

If there are 2 vertices of the K_{n − 4} adjacent to all vertices of S, then ${{C}}_n(G)\supseteq K_2\vee (K_{n-6}+K_{2,2})$. Since K₂ ∨ (K_{n − 6} + K_2,2) is Hamiltonian, then ${{C}}_n(G)$ is Hamiltonian, a contradiction.

If there are 3 vertices of the K_{n − 4} adjacent to all vertices of S, then ${{C}}_n(G)\supseteq K_3\vee (K_{n-7}+2K_2)$. Since K₃ ∨ (K_{n − 7} + 2K₂) is Hamiltonian, then ${{C}}_n(G)$ is Hamiltonian, a contradiction.

If there are 4 vertices of the K_{n − 4} adjacent to all vertices of S, then ${{C}}_n(G)\supseteq K_4\vee (K_{n-8}+4K_1)$. Since K₄ ∨ (K_{n − 8} + 4K₁) is Hamiltonian, then ${{C}}_n(G)$ is Hamiltonian, a contradiction.

If there are more than 4 vertices of the K_{n − 4} adjacent to all vertices of S, ${{C}}_n(G)\supseteq K_i\vee (K_{n-4-i}+4K_1)(i>4)$. Since K_i ∨ (K_{n − 4 − i} + 4K₁) is Hamiltonian, then ${{C}}_n(G)$is Hamiltonian, a contradiction.

Case 2.1.2 k ≠ 4 and 2n − 3k − 11 = 0.

In this case, we can get 16 ≤ n ≤ 21 because $k<\frac{n}{2}$ and 16 = 8t ≤ n, hence n = 16, k = 7. Then d₇ ≤ 7, d₁₁ ≤ 8, d₁₆ ≤ 15, and we have $156\le {\displaystyle \sum _{i=1}^{16}{d}_i}\le 156$ by (3.1) and (3.2). Thus, ${\displaystyle \sum _{i=1}^{16}{d}_i=156}$, and all inequalities of (3.2) become equality. The corresponding permissible graphic sequence is (7⁷, 8⁴, 15⁵). Then there are no two vertices x, y that are not adjacent such that d(x) + d(y) ≤ ∣V(G)∣ − 5. By Corollary 2.7, we can get G is Hamiltonian, a contradiction.

Case 2.2 (k − 4)(2n − 3k − 11) < 0.

In this case, k ≥ 5 and 2n − 3k − 11 < 0. Since $k<\frac{n}{2}$, we have n ≥ 2k + 1. From these results, we have 4k + 2 ≤ 2n ≤ 3k + 10, that is k ≤ 8. Thus, we have 16 = 8t ≤ n ≤ 17.

When n = 16, we have k = 7 from 4k + 2 ≤ 2n ≤ 3k + 10. At this time 2n − 3k − 11 = 0, which contradicts to 2n − 3k − 11 < 0.

When n = 17, we have k = 8 from 4k + 2 ≤ 2n ≤ 3k + 10. In this case, d₈ ≤ 8, d₁₁ ≤ 8, d₁₇ ≤ 16. We have $180\le {\displaystyle \sum _{i=1}^{17}{d}_i\le 184}$ by (3.1) and (3.2). Then, ${\displaystyle \sum _{i=1}^{17}{d}_i=184}$ or ${\displaystyle \sum _{i=1}^{17}{d}_i=182}$ or ${\displaystyle \sum _{i=1}^{17}{d}_i=180}$.

When ${\displaystyle \sum _{i=1}^{17}{d}_i=184}$ or ${\displaystyle \sum _{i=1}^{17}{d}_i=182}$. The corresponding permissible degree sequence and its properties are as follows in Table 1. By Corollary 2.7, we can get G is Hamiltonian, a contradiction.

TABLE 1

Table 1. The degree sequence of G and its properties.

When ${\displaystyle \sum _{i=1}^{17}{d}_i=180}$, The corresponding permissible degree sequence and its properties are shown in the follows in Table 2.

TABLE 2

Table 2. The degree sequence of G and its properties.

From Table 2, we can find that:

(1) for degree sequence (4,8¹⁰,16⁶) and (5,7,8⁹,16⁶). They are not graphic, a contradiction.

(2) for degree sequence (6²,8⁹,16⁶). If the corresponding graphs are not Hamiltonian, there must exist two non-adjacent 6-degree vertices by Corollary 2.7, and the corresponding graphs are isomorphic to K₆ ∨ (C₉ + 2K₁) or K₆ ∨ (C₄ + C₅ + 2K₁) or K₆ ∨ (C₆ + K₃ + 2K₁). We can find these graphs are Hamiltonian, a contraction.

(3) the other degree sequence except (4,8¹⁰,16⁶), (5,7,8⁹,16⁶), and (6²,8⁹,16⁶) in Table 2, there are no two vertices x, y that are not adjacent such that d(x) + d(y) ≤ ∣V(G)∣ − 5. By Corollary 2.7, we can get that G is Hamiltonian, a contradiction.

Case 3 t = 3.

In this case, (k − 6)(2n − 3k − 16) ≤ 0. By Lemma 2.5, δ(G) ≥ 2t = 6. Since δ ≤ d_k ≤ k, then k ≥ 6.

Case 3.1 (k − 6)(2n − 3k − 16) = 0, i.e., k − 6 = 0 or k ≠ 6 and 2n − 3k − 16 = 0.

Case 3.1.1 k = 6.

In this case, G is a graph with d₆ ≤ 6, d_{n − 3} ≤ n − 7, d_n ≤ n − 1. We have $(n-6)(n-7)+54\le {\displaystyle \sum _{i=1}^n{d}_i\le (n-6)(n-7)+54}$ by (3.1) and (3.2), thus ${\displaystyle \sum _{i=1}^n{d}_i=(n-6)(n-7)+54}$. During this time, we have the corresponding permissible degree sequence of G is (6⁶, (n − 7)^{n − 9}, (n − 1)³), and we have n > 9t = 27 because t = 3. Let S is the set containing six 6 − degree vertices of G.

By Lemma 2.8, ${{C}}_n(G)$ is also not Hamiltonian. According to the definition of ${{C}}_n(G)$, all points except the vertices of S form a complete graph K_{n − 6} in ${{C}}_n(G)$. Let us discuss the graph ${{C}}_n(G)$. If one vertex of the K_{n − 6} is adjacent to one vertex of S, it must be adjacent to all vertices of S. Moreover, there are at least 2 vertices of the K_{n − 6} adjacent to all vertices of S because there exist two non-adjacent vertices in S by Corollary 2.7.

If there are 2 vertices of the K_{n − 6} adjacent to all vertices of S, then ${{C}}_n(G)\supseteq K_2\vee (K_{n-8}+C_6)$. Since K₂ ∨ (K_{n − 8} + C₆) is Hamiltonian, then ${{C}}_n(G)$ is Hamiltonian, a contradiction.

If there are 3 vertices of the K_{n − 6} adjacent to all vertices of S, then ${{C}}_n(G)\supseteq K_3\vee (K_{n-9}+C_6)$. Since K₃ ∨ (K_{n − 9} + C₆) is Hamiltonian, then ${{C}}_n(G)$ is Hamiltonian, a contradiction.

If there are 4 vertices of the K_{n − 6} adjacent to all vertices of S, then ${{C}}_n(G)\supseteq K_4\vee (K_{n-10}+C_6)$ or ${{C}}_n(G)\supseteq K_4\vee (K_{n-10}+2C_3)$. Since K₄ ∨ (K_{n − 10} + C₆) and K₄ ∨ (K_{n − 10} + 2C₃) are Hamiltonian, then ${{C}}_n(G)$ is Hamiltonian, a contradiction.

If there are 5 vertices of the K_{n − 6} adjacent to all vertices of S, then ${{C}}_n(G)\supseteq K_5\vee (K_{n-11}+3K_2)$. Since K₅ ∨ (K_{n − 11} + 3K₂) is Hamiltonian, then ${{C}}_n(G)$ is Hamiltonian, a contradiction.

If there are 6 vertices of the K_{n − 6} adjacent to all vertices of S, then ${{C}}_n(G)\supseteq K_6\vee (K_{n-12}+6K_1)$, so C_n(G) is Hamiltonian, a contradiction.

If there are more than 6 vertices of the K_{n − 6} adjacent to all vertices of S, ${{C}}_n(G)\supseteq K_i\vee (K_{n-6-i}+6K_1)(i>6)$. Since K_i ∨ (K_{n − 6 − i} + 6K₁)(i > 6) is Hamiltonian, then ${{C}}_n(G)$is Hamiltonian, a contradiction.

Case 3.1.2 k ≠ 6 and 2n − 3k − 16 = 0.

In this case, we can get 28 ≤ n ≤ 31 because $k<\frac{n}{2}$ and n > 9t = 27. Since 2n − 3k − 16 = 0, then n = 29, k = 14. So d₁₄ ≤ 14, d₁₈ ≤ 14, d₂₉ ≤ 28. We have $560\le {\displaystyle \sum _{i=1}^{29}{d}_i\le 560}$ by (3.1) and (3.2), thus ${\displaystyle \sum _{i=1}^{29}{d}_i=560}$. The corresponding permissible degree sequence is (14¹⁸, 28¹¹). For this degree sequence, we can get that there are no two vertices x, y that are not adjacent such that d(x) + d(y) ≤∣ V(G) ∣ − 5. By Corollary 2.7, we can get G is Hamiltonian, a contradiction.

Case 3.2 (k − 6)(2n − 3k − 16) < 0.

In this case, k ≥ 7 and 2n − 3k − 16 < 0. Since $k<\frac{n}{2}$, we have n ≥ 2k + 1. From these results, we have 4k + 2 ≤ 2n ≤ 3k + 15, that is k ≤ 13. Thus, we have n ≤ 27. A contradiction with known condition n > 9t = 27. ■

Theorem 3.2 Let G be a t − tough(t ∈ {1, 2, 3}) and simple connected graph with n vertices and m edges. If

$\mu (G)\ge \sqrt{n^2-4tn-2n+10t^2+2t-1},$

then

(i) G is Hamiltonian when t ∈ {1, 2} and n ≥ 8t.

(ii) G is Hamiltonian when t = 3 and n > 9t.

Proof Suppose that G is not Hamiltonian. Since K_n is Hamiltonian, then G ≠ K_n. By theorem conditions, we can get n ≥ 8 when t ∈ {1, 2, 3}, thus $\tau (K_{1,n-1})\le \frac{1}{n-1}\le \frac{1}{7}<1$. Thus, G ≠ K_{1, n − 1}.

By Lemma 2.2,

$\sqrt{n^2-4tn-2n+10t^2+2t-1}\le \mu (G)<\sqrt{2m-n+1},$

then

$m>\left(\begin{array}{c}n-2t\\ 2\\ \end{array}\right)+3t^2.$

By Theorem 3.1, we get G is Hamiltonian, a contradiction. ■

Theorem 3.3 Let G be a t − tough(t ∈ {1, 2, 3}) and simple connected graph with n vertices and m edges. If

$q(G)\ge \frac{n^2+10t^2-4nt+2t-n-2+(n-1)(n-2)}{n-1},$

then

(i) G is Hamiltonian when t ∈ {1, 2} and n ≥ 8t.

(ii) G is Hamiltonian when t = 3 and n > 9t.

Proof Suppose that G is not a Hamilton graph. Since K_n is Hamiltonian, then G ≠ K_n and G ≠ K_{n − 1} + v. By theorem conditions, we can get n ≥ 8 when t ∈ {1, 2, 3}, thus $\tau (G)\le \frac{1}{n-1}\le \frac{1}{7}<1$. So, G ≠ K_n, G ≠ K_{1, n − 1}.

By Lemma 2.3,

$\begin{array}{l}\frac{n^2+10t^2-4nt+2t-n-2+(n-1)(n-2)}{n-1}\le \\ q(G)<\frac{2m}{n-1}+n-2,\end{array}$

then

$m>\left(\begin{array}{c}n-2t\\ 2\\ \end{array}\right)+3t^2.$

By Theorem 3.1, we get G is Hamiltonian, a contradiction.

■

4. Concluding remarks

We suggest the following general problems.

Problem 1 Let G be a t − tough(t ∈ {1, 2, 3}) and simple connected graph with n vertices and m edges. If

$m\ge \left(\begin{array}{c}n-2t\\ 2\\ \end{array}\right)+3t^2,$

then

(i) when t ∈ {1, 2} and n ≥ 8t, G is pancyclic.

(ii) when t = 3 and n > 9t, G is pancyclic.

Problem 2 Let G be a t − tough(t ∈ {1, 2, 3}) and simple connected graph with n vertices and m edges. If

$\mu (G)\ge \sqrt{n^2-4tn-2n+10t^2+2t-1},$

then

(i) when t ∈ {1, 2} and n ≥ 8t, G is pancyclic.

(ii) when t = 3 and n > 9t, G is pancyclic.

Problem 3 Let G be a t − tough(t ∈ {1, 2, 3}) and simple connected graph with n vertices and m edges. If

$q(G)\ge \frac{n^2+10t^2-4nt+2t-n-2+(n-1)(n-2)}{n-1},$

then

(i) when t ∈ {1, 2} and n ≥ 8t, G is pancyclic.

(ii) when t = 3 and n > 9t, G is pancyclic.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

GY provide writing ideas and master the thesis as a whole. GC contributed significantly to analysis and manuscript preparation. TY performed the data analyses and wrote the manuscript. HX helped perform the analysis with constructive discussions. All authors contributed to the article and approved the submitted version.

Funding

This study was supported by the National Natural Science Foundation of China (No. 11871077), the NSF of Anhui Province (1808085MA04 and 1908085MC62), the NSF of Anhui Provincial Department of Education (KJ2020A0894 and KJ2021A0650), Graduate Scientific Research Project of Anhui Provincial Department of Education (YJS20210515), Research and Innovation Team of Hefei Preschool Education College (KCTD202001), and Graduate offline course graph theory (2021aqnuxskc02).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

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References

Bauer, D., Broersma, H. J., Vanden, J., and Heuvel Veldman, H. J. (1995a). Long cycles in graphs with prescribed toughness and minimum degree. Discrete Math. 141, 1–10. doi: 10.1016/0012-365X(93)E0204-H

CrossRef Full Text | Google Scholar

Bauer, D., Schmeichel, E., and Veldman, H. J. (1991). “Some recent results on long cycles in tough graphs,” in Proceedings of 6th International Conference on the Theory and Applications of Graphs. 113–123. Available online at: https://www.narcis.nl/publication/RecordID/oai%3Aris.utwente.nl%3Apublications%2Faa18e2e5-a533-4466-b771-88afed2d188c

Google Scholar

Bauer, D., Schmeichel, E., and Veldman, H. J. (1995b). “Cycles in tough graphs updating the last 4 years,” in Graph Theory, Combinatorics, and Applications: Proceeding of the Seventh Quadrennial International Conference on the Theory and Applications of Graphs, Vol. 1, 19–34. Available online at: https://www.narcis.nl/publication/RecordID/oai%3Aris.utwente.nl%3Apublications%2F3e216507-2b8f-4b19-9589-d635ae8cd20f

Google Scholar

Bauer, D., Schmeichel, E., and Veldman, H. J. (1999). Progress on tough graphs another four years. Combinatorica 1, 69–88.

Google Scholar

Bauer, D., Broersma, H., and Schmeichel, E. (2006). Toughness in graphs -a survey. Graphs Combinat. 22, 1–35. doi: 10.1007/s00373-006-0649-0

CrossRef Full Text | Google Scholar

Bondy, J. A., and Chvátal, V. (1976). A method in graph theory. Discrete Math. 15, 111–135. doi: 10.1016/0012-365X(76)90078-9

CrossRef Full Text | Google Scholar

Bondy, J. A., and Murty, U. S. R. (1982). Graph Theory with Applications. ISBN: 978-0-333-22694-0.

Google Scholar

Chvátal, V. (1973). Tough graphs and Hamiltonian circuits. Discrete Math. 2, 111–113. doi: 10.1016/0012-365X(73)90138-6

CrossRef Full Text | Google Scholar

Feng, L., Zhang, P., Liu, H., Liu, W., Liu, M., and Hu, Y. (2017). Spectral conditions for some graphical properties. Linear Algebra Appl. 524, 182–198. doi: 10.1016/j.laa.2017.03.006

CrossRef Full Text | Google Scholar

Fiedler, M., and Nikiforov, V. (2010). Spectral radius and Hamiltonicity of graphs. Linear Algebra Appl. 432, 2170–2173. doi: 10.1016/j.laa.2009.01.005

CrossRef Full Text | Google Scholar

Hoàng, C. (1995). Hamiltonian degree conditions for tough graphs. Discrete Math. 142, 121–139. doi: 10.1016/0012-365X(93)E0214-O

CrossRef Full Text | Google Scholar

Huang, X., Das, K. C., and Zhu, S. (2022). Toughness and normalized Laplacian eigenvalues of graphs. Appl. Math. Comput. 425, 127075. doi: 10.1016/j.amc.2022.127075

CrossRef Full Text | Google Scholar

Jung, H. (1978). On maximal circuits in finite graphs. Ann. Discrete Math. 3, 129–144. doi: 10.1016/S0167-5060(08)70503-X

CrossRef Full Text | Google Scholar

Li, B., and Ning, B. (2016). Spectral analogues of Erdős' and MoonMoser's theorems on Hamilton cycles. Linear Multilinear Algebra 64, 2252–2269. doi: 10.1080/03081087.2016.1151854

CrossRef Full Text | Google Scholar

Liu, R., Shiu, W., and Xue, J. (2015). Sufficient spectral conditions on Hamiltonian and traceable graphs. Linear Algebra Appl. 467, 254–266. doi: 10.1016/j.laa.2014.11.017

CrossRef Full Text | Google Scholar

Lu, M., Liu, H., and Tian, F. (2012). Spectral radius and Hamiltonian graphs. Linear Algebra Appl. 437, 1670–1674. doi: 10.1016/j.laa.2012.05.021

CrossRef Full Text | Google Scholar

Yu, G., and Fan, Y. (2013). Spectral conditions for a graph to be hamilton-connected. Appl. Mech. Mater. 336–338, 2329–2334. doi: 10.4028/www.scientific.net/AMM.336-338.2329

CrossRef Full Text | Google Scholar

Yu, G., Fang, Y., Fan, Y., and Cai, G. (2019). Spectral radius and Hamiltonicity of graphs. Discuss. Math. Graph Theory 39, 951–974. doi: 10.7151/dmgt.2119

CrossRef Full Text | Google Scholar

Yuan, H. (1988). A bound on the spectral radius of graphs. Linear Algebra Appl. 108, 135–139. doi: 10.1016/0024-3795(88)90183-8

PubMed Abstract | CrossRef Full Text | Google Scholar

Zhou, B. (2010). Signless laplacian spectral radius and hamiltonicity. Linear Algebra Appl. 432, 566–570. doi: 10.1016/j.laa.2009.09.004

CrossRef Full Text | Google Scholar

Zhou, Q., Wang, L., and Lu, Y. (2018). Some sufficient conditions on Hamiltonian and traceable graphs. Adv. Math. 47, 31–40. doi: 10.1080/03081087.2016.1182463

CrossRef Full Text | Google Scholar