Intersection Of Neighborhoods Research Articles

Hamiltonicity, neighborhood intersections and the partially square graphs

Let G be a graph. The partially square graph G ∗ of G is a graph obtained from G by adding edges uv satisfying the conditions uv∉E(G), and there is some w∈N(u)∩N(v), such that N(w)⊆N(u)∪N(v)∪{u,v}. A non-negative rational sequence (a 1,a 2,…,a k+1) is called an LTW-sequence if the following conditions are satisfied: (1) a 1⩽1; (2) for arbitrary i 1,i 2,…,i h∈{2,3,…,k+1},∑ j=1 hi j⩽k+1 implies ∑ j=1 h(a i j −1)⩽1 . In this paper, we will use the technique of the vertex insertion on l-connected ( l=k,k−1 or k+1,k⩾2) graphs to provide a unified proof for G to be hamiltonian, traceable, 1-hamiltonian or hamiltonian-connected, the sufficient conditions are expressed by weighted sums of the neighborhood intersections in G of independent sets in G ∗ , where the weights are LTW-sequences.

Insertible vertices, neighborhood intersections, and hamiltonicity

AbstractLet G be a simple undirected graph of order n. For an independent set S ⊂ V(G) of k vertices, we define the k neighborhood intersections Si = {v ϵ V(G)\S|N(v) ∩ S| = i}, 1 ≦ i ≦ k, with si = |Si|. Using the concept of insertible vertices and the concept of neighborhood intersections, we prove the following theorem.

Hamiltonian graphs with neighborhood intersections

AbstractIn this paper, k + 1 real numbers c1, c2, ⃛, ck+1 are found such that the following condition is sufficient for a k‐connected graph of order n to be hamiltonian: for each independent vertex set of k + 1 vertices in G.magnified image where Si = {v ≅ V:|N(v) ∩ S| = i} for 0 ≦ i ≦ k + 1. Such a set of k + 1 numbers is called an Hk‐sequence. A sufficient condition for the existence of Hk‐sequences is obtained that generalizes many known results involving sum of degrees, neighborhood unions, and/or neighborhood intersections.

Hamiltonian graphs involving neighborhood intersections

Let G be a graph of order n and α the independence number of G. We show that if G is a 2- connected graph and max { d( u), d( v)}⩾ n/2 for each pair of non-adjacent vertices u, v with 1 ⩽| N( u) ∩ N( v)| ⩾ α − 1, then G is hamiltonian. This result generalizes Fan's (1984) result on hamiltonian graphs.

Hamiltonism, degree sum and neighborhood intersections

We give a sufficient condition for hamiltonism of a 2-connected graph involving the degree sum and the neighborhood intersection of any three independent vertices.

Minimal first countable spaces

A topological space isE0(resp.E1) provided every point is the countable intersection of neighborhoods (resp. closed neighborhoods). Fori= 0 andi= 1, characterizations of minimalEi. spaces (Ei. spaces with no strictly coarserEi. topology) andEi-closed spaces (Ei. spaces which are closed in everyEi. space containing them) are given; for example, the properties of minimalEi. and minimal first countableTi+1are shown to be equivalent. MinimalE0spaces are characterized as countable spaces with the cofinite topology, and minimalE1spaces are characterized asE1-closed and semiregular spaces.E0-closed spaces are shown to be precisely the finite discrete spaces.