**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**3

# cycle Related Publications

##### 3 Mutually Independent Hamiltonian Cycles of Cn x Cn

**Authors:**
Kai-Siou Wu,
Justie Su-Tzu Juan

**Abstract:**

In a graph G, a cycle is Hamiltonian cycle if it contain all vertices of G. Two Hamiltonian cycles C_1 = ⟨u_0, u_1, u_2, ..., u_{n−1}, u_0⟩ and C_2 = ⟨v_0, v_1, v_2, ..., v_{n−1}, v_0⟩ in G are independent if u_0 = v_0, u_i = ̸ v_i for all 1 ≤ i ≤ n−1. In G, a set of Hamiltonian cycles C = {C_1, C_2, ..., C_k} is mutually independent if any two Hamiltonian cycles of C are independent. The mutually independent Hamiltonicity IHC(G), = k means there exist a maximum integer k such that there exists k-mutually independent Hamiltonian cycles start from any vertex of G. In this paper, we prove that IHC(C_n × C_n) = 4, for n ≥ 3.

**Keywords:**
independent,
Hamiltonian,
cycle,
Cartesian product,
mutually independent Hamiltonicity

##### 2 Neighbors of Indefinite Binary Quadratic Forms

**Authors:**
Ahmet Tekcan

**Abstract:**

**Keywords:**
cycle,
Quadratic form,
indefinite form,
proper cycle,
right neighbor,
left neighbor

##### 1 The Partial Non-combinatorially Symmetric N10 -Matrix Completion Problem

**Authors:**
Gu-Fang Mou,
Ting-Zhu Huang

**Abstract:**

An n×n matrix is called an N1 0 -matrix if all principal minors are non-positive and each entry is non-positive. In this paper, we study the partial non-combinatorially symmetric N1 0 -matrix completion problems if the graph of its specified entries is a transitive tournament or a double cycle. In general, these digraphs do not have N1 0 -completion. Therefore, we have given sufficient conditions that guarantee the existence of the N1 0 -completion for these digraphs.

**Keywords:**
digraph,
matrix completion,
cycle,
N10 -matrix,
non-combinatorially symmetric