Neighborhood conditions for graphs with induced claws

Hamiltonian graphs involving neighborhood conditions

Let G be a graph on n vertices, δ and α be the minimum degree and independence number of G, respectively. In this paper we prove that if G is a 2-connected graph and |N(x) ∪ N(y)| ≥ n − δ − 1 for each pair of nonadjacent vertices x, y with 1 ≤ |N(x) ∩ N(y)| ≤ α − 1, then G is hamiltonian or G ∈ {Gn−1 2 ,Kn+1 2 ,K∗ 2 ∨ 3Kn−2 3 } where K∗ 2 and Gn−1 2 are subgraphs on 2 and n−1 2 vertices respect...

Neighborhood conditions for balanced independent sets in bipartite graphs

Let G be a balanced bipartite graph of order 2n and minimum degree (G) 3. If for every balanced independent set S of four vertices in G the neighborhood union jN(S)j > n then G is traceable, the circumference is at least 2n ? 2 and G contains a 2-factor (with one exceptional graph for the latter statement). If jN(S)j > n + 2 then G is hamiltonian.

Neighborhood Union Conditions for Hamiltonicity of P 3-Dominated Graphs

A graph G is called P3-dominated (P3D) if it satisfies J(x, y) ∪ J ′(x, y) 6= ∅ for every pair (x, y) of vertices at distance 2, where J(x, y) = {u|u ∈ N(x) ∩N(y), N [u] ⊆ N [x]∪N [y]} and J ′(x, y) = {u|u ∈ N(x)∩N(y)| if v ∈ N(u)\(N [x]∪N [y]), then (N(u)∪ N(x) ∪ N(y))\{x, y, v} ⊆ N(v)} for x, y ∈ V (G) at distance 2}. For a noncomplete graph G, the number NC is defined as NC = min{|N(x) ∪ N(y...

Distance-regular graphs without 4-claws

We determine the distance-regular graphs with diameter at least 3 and c2 ≥ 2 but without induced K1,4-subgraphs.

Clustering with neighborhood graphs