The Connected Domination Number and Some Hamiltonian Properties of Graphs

1. Introduction

We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow those in [1]. For a graph $G=\left(V\right(G),E(G\left)\right)$, we use n and e to denote its order $\left|V\right(G\left)\right|$ and size $\left|E\right(G\left)\right|$, respectively. The minimum degree and maximum degree of G are denoted by $\delta \left(G\right)$ and $\Delta \left(G\right)$, respectively. The neighborhood of a vertex u in G is denoted by $N_G\left(u\right)$. If a vertex $u\in V-V_1$, where $V_1$ is a subset of the vertex set V, we define $d_{G\left[V_1\right]}\left(u\right)$ as $|N_G\left(u\right)\cap V_1|$, where $G\left[V_1\right]$ is a subgraph of G which is induced by $V_1$. A subset $V_1$ of the vertex set $V\left(G\right)$ of G is called an independent set if no two vertices in $V_1$ are adjacent in G. A maximum independent set in a graph G is an independent set of largest possible size. The independence number, denoted $\alpha \left(G\right)$, of a graph G is the cardinality of a maximum independent set in G. A subset D of $V\left(G\right)$ of G is called a dominating set if every vertex v in G is either in the set D or is adjacent to a vertex in D. A connected dominating set of G is a subset of V that is a dominating set and induces a connected subgraph of G. The connected domination number of G, denoted ${\gamma }_c\left(G\right)$, is the minimum size of all connected dominating sets of G. Let $A\left(G\right)$ be the adjacency matrix of G. The Lapalcian spectral radius of G, denoted ${\mu }_1\left(G\right)$, is defined as the largest eigenvalue of $L\left(G\right):=D\left(G\right)-A\left(G\right)$, where $D\left(G\right)$ is a diagonal matrix whose entries are the degrees of vertices in G. The signless Lapalcian spectral radius of G, denoted $q_1\left(G\right)$, is defined as the largest eigenvalue of $Q\left(G\right):=D\left(G\right)+A\left(G\right)$, where $D\left(G\right)$ is a diagonal matrix whose entries are the degrees of vertices in G. A cycle C in a graph G is called a Hamiltonian cycle of G if C contains all the vertices of G. A graph G is called Hamiltonian if G has a Hamiltonian cycle. A path P in a graph G is called a Hamiltonian path of G if P contains all the vertices of G. A graph G is called traceable if G has a Hamiltonian path. In this note, we present sufficient conditions based on the connected domination number for some Hamiltonian properties of graphs. The main results are as follows.

Theorem 1.

Let G be a k-connected graph ($k\ge 2$) graph of order $n\ge 3$ and size e. If ${\gamma }_c+{\displaystyle \frac{n\delta }{2(n-k-1)}}\ge n$, then G is Hamiltonian.

Theorem 2.

Let G be a k-connected graph ($k\ge 1$) graph of order $n\ge 9$ and size e. If ${\gamma }_c+{\displaystyle \frac{n\delta }{2(n-k-2)}}\ge n$, then G is traceable.

2. Lemmas

We need the following results as lemmas to prove Theorem 1 and Theorem 2. Lemma 1 and Lemma 2 below are from [2].

Lemma 1. 

Let G be a k-connected graph of order $n\ge 3$. If $\alpha \le k$, then G is Hamiltonian.

Lemma 2. 

Let G be a k-connected graph of order n. If $\alpha \le k+1$, then G is traceable.

Lemma 3 below is from [5]. See also Proposition $10.2$ on Page 272 in [4].

Lemma 3. 

Let G be a connected graph of order n. Then ${\gamma }_c+\Delta \le n$.

Lemma 4 below is from Theorem $4.6$ on Page 163 in [3].

Lemma 4. 

Let G be a connected graph. Then $q_1\le 2\Delta$ with equality if and only if G is regular.

Lemma 5 below is from Lemma $2.1$ on Page 34 in [8].

Lemma 5. 

Let G be a connected. Then ${\mu }_1\le q_1$ with equality if and only if G is a bipartite graph.

Lemma 6 below follows from Proof of [4] in Theorem $1.1$ on Page 584 in [6].

Lemma 6. 

Let G be a connected graph and I is an independent set with $\left|I\right|=\alpha$. Then ${\mu }_1\ge \left({\sum }_{y\in I}d\left(y\right)\right){\displaystyle \frac{n}{\alpha (n-\alpha )}}$.

Lemma 7 below is from [7].

Lemma 7. 

Let G be a balanced bipartite graph of order $2n$ with bipartition (A, B). If $d\left(x\right)+d\left(y\right)\ge n+1$ for any $x\in A$ and any $y\in B$ with $xy\notin E$, then G is Hamiltonian.

3. Proofs

Proof of Theorem 1. 

Let G be a k-connected ($k\ge 2$) graph with $n\ge 3$ vertices and e edges satisfying the conditions in Theorem 1. Suppose G is not Hamiltonian. Lemma 1 implies that $\alpha \ge k+1$. Let I be any maximum independent set in G. Then $\left|I\right|=\alpha <n$. From Lemmas 3, 4, 5, and 6, we have that

$${\gamma }_c+{\displaystyle \frac{n\delta }{2(n-k-1)}}\ge n\ge {\gamma }_c+\Delta \ge {\gamma }_c+q_1/2$$

$$\ge {\gamma }_c+{\mu }_1/2\ge {\gamma }_c+\left(\sum _{y\in I}d\left(y\right)\right){\displaystyle \frac{n}{2\alpha (n-\alpha )}}$$

$$\ge {\gamma }_c+{\displaystyle \frac{\alpha \delta n}{2\alpha (n-\alpha )}}={\gamma }_c+{\displaystyle \frac{n\delta }{2(n-\alpha )}}\ge {\gamma }_c+{\displaystyle \frac{n\delta }{2(n-k-1)}}.$$

Thus $2\Delta =q_1$, $q_1={\mu }_1$, and $\alpha =k+1$. Therefore G is a k-connected ($k\ge 2$) regular bipartite graph. Suppose the partition sets of G are P and Q. Then $\delta \left|P\right|=\left|E\right(I,V-I\left)\right|=\delta \left|Q\right|$. Thus $\left|P\right|=\left|Q\right|$. Notice that both P and Q are maximum independent sets in G. Therefore $P|=|Q|=\alpha =k+1$. From Lemma 7, we have that G is Hamiltonian, a contradiction. □

Proof of Theorem 2. 

Let G be a k-connected ($k\ge 1$) graph with $n\ge 9$ vertices and e edges satisfying the conditions in Theorem 2. Suppose G is not traceable. Lemma 2 implies that $\alpha \ge k+2$. Let I be any maximum independent set in G. Then $\left|I\right|=\alpha <n$. From Lemmas 3, 4, 5, and 6, we have that

$${\gamma }_c+{\displaystyle \frac{n\delta }{2(n-k-2)}}\ge n\ge {\gamma }_c+\Delta \ge {\gamma }_c+q_1/2$$

$$\ge {\gamma }_c+{\mu }_1/2\ge {\gamma }_c+\left(\sum _{y\in I}d\left(y\right)\right){\displaystyle \frac{n}{2\alpha (n-\alpha )}}$$

$$\ge {\gamma }_c+{\displaystyle \frac{\alpha \delta n}{2\alpha (n-\alpha )}}={\gamma }_c+{\displaystyle \frac{n\delta }{2(n-\alpha )}}\ge {\gamma }_c+{\displaystyle \frac{n\delta }{2(n-k-2)}}.$$

Thus $2\Delta =q_1$, $q_1={\mu }_1$, and $\alpha =k+2$. Therefore G is a k-connected ($k\ge 1$) regular bipartite graph. Suppose the partition sets of G are P and Q. Then $\delta \left|P\right|=\left|E\right(I,V-I\left)\right|=\delta \left|Q\right|$. Thus $\left|P\right|=\left|Q\right|$. Notice that both P and Q are maximum independent sets in G. Therefore $P|=|Q|=\alpha =k+2$. Since $n=2k+4\ge 9$, we have that $k\ge 3$. From Lemma 7, we have that G is traceable, a contradiction. □

Remark. 

In the Proof of Theorems 1 and 2, we establish an inequality on the $\alpha$ and ${\gamma }_c$ for a connected graph. Namely, ${\gamma }_c+{\displaystyle \frac{n\delta }{2(n-\alpha )}}\le n$. Notice further that for some connected graphs the equality in the less than or equal to relationship is achievable. For instance, if G is an even cycle of $C_{2s}$ with $s\ge 2$. Then $n=2s$, ${\gamma }_c=2s-2$, $\delta =2$, $\alpha ={\displaystyle \frac{n}{2}}$, and ${\gamma }_c+{\displaystyle \frac{n\delta }{2(n-\alpha )}}=n$.