An implicit degree condition for k-connected 2-heavy graphs to be hamiltonian

Highlights

• We give a sufficient condition for a k-connected 2-heavy graph to be hamiltonian.

• Our result generalizes a previous theorem of Chen, Egawa, Liu and Saito.

• We give an example to show that the low bound of implicit degree is best possible.

• We give an example to show that our result is stronger than the result of Chen et. al.

Abstract

Let $id(v)$ denote the implicit degree of a vertex v in a graph G. An independent set S of G is said to be essential if S contains a pair of vertices at distance 2 in G. A graph G on $n\ge 3$ vertices is called 2-heavy if there exist at least two end-vertices of every induced claw having implicit degree at least $n/2$. In this paper, we prove that: Let G be a k-connected ($k\ge 2$) 2-heavy graph on $n\ge 3$ vertices. If $\max \{id(v):v\in S\}\ge n/2$ for every essential independent set S of order k, then G is hamiltonian.

Introduction

In this paper, we consider only undirected and finite graphs without loops or multiple edges. Notation and terminology not defined here can be found in [3].

Let G be a graph and H be a subgraph of G. For a vertex $u\in V(G)$, we denote the neighborhood of u in H by $N_H(u)$, the degree of u in H by $d_H(u)=|N_H(u)|$, the distance between u and v in H by $d_H(u,v)$. If $H=G$, we can use $N(u)$, $d(u)$ and $d(u,v)$ in place of $N_G(u)$, $d_G(u)$ and $d_G(u,v)$, respectively.

Let $G_1,\dots ,G_k$ be k vertex disjoint graphs. The union of $G_1,\dots ,G_k$, denoted by $G_1\cup \dots \cup G_k$, is the graph with vertex set ${\cup }_{i=1}^{k}V(G_i)$ and edge set ${\cup }_{i=1}^{k}E(G_i)$. We use kQ instead of $G_1\cup \dots \cup G_k$ if each $G_i$ is isomorphic to Q. The join of $G_1,\dots ,G_k$, denoted by $G_1\vee \dots \vee G_k$, is the graph obtained from ${\cup }_{i=1}^k{G}_i$ by joining each vertex of $G_i$ to each vertex of $G_j$ for $i\ne j$.

An induced subgraph of G with vertex set $\{u,v,w,x\}$ and edge set $\{uv,uw,ux\}$ is called a claw of G, with center u and end-vertices $v,w,x$. G is called claw-free if G does not contain an induced subgraph isomorphic to a claw. A vertex v of G is called heavy if $d(v)\ge |V(G)|/2$. Following [4], a claw of G is called 2-heavy if at least two of its end vertices are heavy. A graph is called 2-heavy if all its claws are 2-heavy.

A cycle in a graph G is called a Hamilton cycle if it contains all vertices of G. G is called hamiltonian if it contains a Hamilton cycle. There are many sufficient conditions for a graph to be hamiltonian, and many of them involve some sort of degree conditions. Dirac's theorem is a classical result among them.

Theorem 1

([10]) Let G be a graph on $n\ge 3$ vertices. If $d(u)\ge n/2$ for every vertex u in G, then G is hamiltonian.

In 1984, Fan [11] proved that only the pairs of vertices that are at distance 2 are essential in Theorem 1 and the condition can be replaced by a weaker one.

Theorem 2

([11]) Let G be a 2-connected graph on $n\ge 3$ vertices. If $\max \{d(u),d(v)\}\ge n/2$ for every pair of vertices u and v at distance 2 in G, then G is hamiltonian.

In [8], the authors proved that a k-connected graph is hamiltonian if it contains no independent set S with $|S|=k+1$. In 1980, by considering the degree sums of vertices in independent sets with $k+1$ vertices, Bondy [2] obtained a sufficient condition for a k-connected graph to be hamiltonian.

Theorem 3

([2]) Let G be a k-connected graph on $n\ge 3$ vertices. If ${\sum }_{u\in S}d(u)>(k+1)(n-1)/2$ for each independent set S of order $k+1$, then G is hamiltonian.

Let G be a graph and S be a subset of $V(G)$. S is said to be an essential independent set if S is an independent set and contains two distinct vertices u and v at distance 2 in G. Chen et al. [9] gave a Fan-type sufficient condition for a k-connected graph to be hamiltonian by using degrees of vertices in essential independent set of order k.

Theorem 4

([9]) Let G be a k-connected graph on $n\ge 3$ vertices ($k\ge 2$). If $\max \{d(u):u\in S\}\ge n/2$ for every essential independent set S with k vertices, then G is hamiltonian.

Moreover, the authors in [9] gave an example to show that the condition “$\max \{d(u):u\in S\}\ge n/2$ for every essential independent set S with k vertices” in Theorem 4 can not be exchanged by “$\max \{d(u):u\in S\}\ge n/2$ for every essential independent set S with $k+1$ vertices”.

In order to generalize some classical results about hamiltonian problems, Zhu, Li and Deng [14] gave the definition of implicit degree of a vertex. We use $N_2(v)$ to denote the vertices which are at distance 2 from v in G.

Definition 1

([14]) Let v be a vertex of a graph G and $d(v)=l+1$. Set $M_2=\max \{d(u):u\in N_2(v)\}$ and $m_2=\min \{d(u):u\in N_2(v)\}$. If $N_2(v)\ne \varnothing$ and $d(v)\ge 2$, then let $d_1\le d_2\le d_3\le \dots \le d_l\le d_{l+1}\le \dots$ be the degree sequence of vertices of $N(v)\cup N_2(v)$. Define

$$d^{⁎}(v)=\{\begin{array}{cc}{m}_2,\hfill & \text{ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} if \hspace{0.17em}\hspace{0.17em}\hspace{0.17em} }{d}_l<m_2\text{;}\hfill \\ d_{l+1},\hfill & \text{ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} if \hspace{0.17em}\hspace{0.17em}\hspace{0.17em} }{d}_{l+1}>M_2\text{;}\hfill \\ d_l,\hfill & \text{ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} otherwise.}\hfill \end{array}$$

Then the implicit degree of v is defined as $id(v)=\max \{d(v),d^{⁎}(v)\}$. If $N_2(v)=\varnothing$ or $d(v)\le 1$, then define $id(v)=d(v)$.

Clearly, $id(v)\ge d(v)$ for every vertex v from the definition of implicit degree. The authors [14] gave a sufficient condition for a 2-connected graph to be hamiltonian by considering the relationship between the order of the graph and the implicit degree sum of nonadjacent vertices.

Theorem 5

([14]) Let G be a 2-connected graph on $n\ge 3$ vertices. If $id(u)+id(v)\ge n$ for every pair of nonadjacent vertices u and v in G, then G is hamiltonian.

For more results using implicit degree conditions, we refer to [6], [5], [7] and [12]. Recently, Li, Ning and Cai [13] gave a sufficient condition for a k-connected graph to be hamiltonian by replacing the degree sum in Theorem 3 by implicit degree sum.

Theorem 6

([13]) Let G be a k-connected graph on $n\ge 3$ vertices. If ${\sum }_{u\in S}id(u)>(k+1)(n-1)/2$ for each independent set S with $k+1$ vertices, then G is hamiltonian.

The main purpose of this paper is to generalize Theorem 4 in the way as Theorem 3 to Theorem 6 and give an implicit Fan-type sufficient condition for a k-connected graph to be hamiltonian by using implicit degrees of vertices in essential independent sets. We will prove the following result.

Theorem 7

Let G be a k-connected 2-heavy graph on $n\ge 3$ vertices ($k\ge 2$). If $\max \{id(v):v\in S\}\ge n/2$ for every essential independent set S with k vertices, then G is hamiltonian.

The proof of Theorem 7 will be given in next section. Now we present the following remarks. The first one shows the sharpness of Theorem 7 and the other one shows a graph which does not satisfy the condition of Theorem 4 but it can be easily verified to be hamiltonian by using Theorem 7.

Remark 1

Let $G=K_k\vee (k+1)K_1$. It is easy to check that G has $n=2k+1$ vertices and G is a k-connected nonhamiltonian graph. And for each essential independent set S with k vertices, $\max \{id(x):x\in S\}=(n-1)/2$ and each induced claw with two vertices of degree $(n-1)/2$. This implies that the lower bound $n/2$ of Theorem 7 is best possible.

Remark 2

Let $k\ge 2$ be a nonnegative integer. Set $V(K_{k^2})=\{x_1,x_2,\dots ,x_{k^2}\}$ and $V(k{K}_1)=\{y_1,y_2,\dots ,y_k\}$. We choose a graph G with $V(G)=V(K_{k^2}\cup k{K}_1)$ and $E(G)=E(K_{k^2})\cup \{y_i{x}_{k(i-1)+j}:i,j=1,2,\dots ,k\}$ (see Fig. 1). It is easy to see that G has $n=k^2+k$ vertices and G is a hamiltonian graph not satisfying the conditions of Theorem 4. But since $id(x)=k^2>n/2$ for each $x\in V(G)$, G satisfies the conditions of Theorem 7. This shows that our result in Theorem 7 does strengthen the result in Theorem 4.