The clique number and some Hamiltonian properties of graphs

Abstract A graph is said to be Hamiltonian (respectively, traceable) if it has a Hamiltonian cycle (respectively, Hamiltonian path), where a Hamiltonian cycle (respectively, Hamiltonian path) is a cycle (respectively, path) containing all the vertices of the graph. In this short note, sufficient conditions involving the clique number for the Hamiltonian and traceable graphs are presented.


Introduction and statements of the main results
Throughout this note, only finite undirected graphs without loops or multiple edges are considered. The notation and terminology used in this note, but not defined here, can be found in the book [1]. For a graph G = (V, E), its order is denoted by n, that is n = |V |. Denote by δ(G), ω(G), and α(G) the minimum degree, the clique number, and the independence number of a graph G, respectively. For each positive integer r ≤ α(G), define 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. Recall the following well-known results obtained by Chvátal and Erdős in [2]. Theorem 1.1. Let G be a k-connected graph of order n ≥ 3. If α ≤ k, then G is Hamiltonian.
From Theorem 1.1, one can see that it is reasonable to find sufficient conditions for the Hamiltonicity of k-connected graphs when α ≥ k + 1 and k ≥ 2. Also, from Theorem 1.2, one can see that it is reasonable to find sufficient conditions for the traceability of k-connected graphs when α ≥ k + 2 and k ≥ 1. In this short note, sufficient conditions involving the clique number, for the Hamiltonicity of k-connected (k ≥ 2) graphs with the constraint α ≥ k + 1 are presented. Sufficient conditions involving the clique number, for the traceability of k-connected (k ≥ 1) graphs with the constraint α ≥ k + 2 are also presented. The main results of this note are as follows.
Theorem 1.4. Let G be a k-connected graph of order n with α ≥ k + 2 ≥ 3. If

Proofs of Theorems 1.3 and 1.4
In order to prove Theorem 1.3 and Theorem 1.4, we need the following result obtained in [3].
. , x n ) be any n-vector with Proof of Theorem 1.3. Let G be a graph satisfying the conditions of Theorem 1.3. Suppose that G is not Hamiltonian.
Define an n-vector (x 1 , x 2 , . . . , x n ) as follows. For each i with 1 ≤ i ≤ α, set and for each i with α + 1 ≤ i ≤ n, set .
Then x 1 + x 2 + · · · + x n = 1 and x i ≥ 0 for each i, 1 ≤ i ≤ n. By applying Lemma 2.1 on the graph G with the n-vector (x 1 , x 2 , . . . , x n ) defined above, one has Therefore, all the above inequalities become equalities. This implies that vivj ∈E, vi∈V −I, vj ∈V −I