SUFFICIENT CONDITIONS FOR SOME HAMILTONIAN PROPERTIES AND K-CONNECTIVITY OF GRAPHS

Abstract

For a connected graph G = (V, E), its inverse degree is defined as $\sum_{{\upsilon}{\in}{V}}^{}\frac{1}{d(\upsilon)}$. Using an upper bound for the inverse degree of a graph obtained by Cioabă in [4], we in this note present sufficient conditions for some Hamiltonian properties and k-connectivity of a graph.

1. Introduction

We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [3]. For a graph G = (V, E), we use n and e to denote its order |V| and size |E|, respectively. We use δ = d1 ≤ d2 ≤ · · · ≤ dn = Δ to denote the degree sequence of G. If G is connected, we define its inverse degree as 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. A graph G is called Hamiltonconnected if for each pair of vertices in G there is a Hamiltonian path between them. In this note, we will use an upper bound for the inverse degree of a graph obtained by Cioabă in [4] to present sufficient conditions for Hamiltonian, traceable, Hamilton-connected, and k-connected graphs.

 

2. Main results

The main results of this paper are as follows.

Theorem 2.1. Let G be a connected graph of order n ≥ 3 and size e. If

then G is Hamiltonian.

Theorem 2.2. Let G be a connected graph of order n ≥ 4 and size e. If

then G is traceable.

Theorem 2.3. Let G be a connected graph of order n ≥ 3 and size e. If

then G is Hamilton-connected.

Theorem 2.4. Let G be a connected graph of order n ≥ k + 1 ≥ 3 and size e. If

then G is k-connected.

 

3. Lemmas

In order to prove the theorems above, we need the following results as our lemmas.

Lemma 3.1 ([1]). Let G be a graph of order n ≥ 3 with degree sequence d1 ≤ d2 ≤ · · · ≤ dn. If

then G is Hamiltonian.

Lemma 3.2 ([1]). Let G be a graph of order n ≥ 2 with degree sequence d1 ≤ d2 ≤ · · · ≤ dn. If

then G is traceable.

Lemma 3.3 ([1]). Let G be a graph of order n ≥ 3 with degree sequence d1 ≤ d2 ≤ · · · ≤ dn. If

then G is Hamilton-connected.

Lemma 3.4 ([2]). Let G be a graph of order n ≥ 2 with degree sequence d1 ≤ d2 ≤ · · · ≤ dn and let 1 ≤ k ≤ n − 1. If

then G is k-connected.

Lemma 3.5 ([4]). Let G be a connected graph of order n and size e. Then

Notice that Lemma 3.1 is Corollary 3 on Page 209 in [1] or Theorem 4.5 on Page 57 in [3], Lemma 3.2 is Corollary 6 on Page 210 in [1], Lemma 3.3 is Theorem 12 on Page 218 in [1], Lemma 3.4 is the Corollary on Page 163 in [2], and Lemma 3.5 is from Theorem 9 on Page 1963 in [4].

 

4. Proofs

Proof of Theorem 2.1. Let G be a graph satisfying the conditions in Theorem 2.1. Suppose that G is not Hamiltonian. Then, from Lemma 3.1, there exists an integer k such that and dn−k ≤ n − k − 1. Obviously, k ≥ 1. Therefore, from Lemma 3.5, we have that

a contradiction. This completes the proof of Theorem 2.1. □

Proof of Theorem 2.2. Let G be a graph satisfying the conditions in Theorem 2.2. Suppose that G is not traceable. Then, from Lemma 3.2, there exists an integer k such that and dn+1−k ≤ n − k − 1. Obviously, k ≥ 2. Therefore, from Lemma 3.5, we have that

a contradiction. This completes the proof of Theorem 2.2. □

Proof of Theorem 2.3. Let G be a graph satisfying the conditions in Theorem 2.3. Suppose that G is not Hamilton-connected. Then, from Lemma 3.3, there exists an integer k such that and dn-k ≤ n − k. Therefore, from Lemma 3.5, we have that

a contradiction. This completes the proof of Theorem 2.3. □

Proof of Theorem 2.4. Let G be a graph satisfying the conditions in Theorem 2.4. Suppose that G is not k-connected. Then, from Lemma 3.4, there exists an integer j such that and dn−k+1 ≤ n − j − 1. Therefore, from Lemma 3.5, we have that

a contradiction. This completes the proof of Theorem 2.3. □

Proof of Theorem 2.4. Let G be a graph satisfying the conditions in Theorem 2.4. Suppose that G is not k-connected. Then, from Lemma 3.4, there exists an integer j such that and dn−k+1 ≤ n − j − 1. Therefore, from Lemma 3.5, we have that

a contradiction. This completes the proof of Theorem 2.4. □