On graphs satisfying a local ore-type condition

Abstract

For an integer i, a graph is called an Li‐graph if, for each triple of vertices u, v, w with d(u, v) = 2 and w (element of) N(u) (intersection) N(v), d(u) + d(v) ≥ | N(u) (union) N(v) (union) N(w)| —i. Asratian and Khachatrian proved that connected Lo‐graphs of order at least 3 are hamiltonian, thus improving Ore's Theorem. All K1,3‐free graphs are L1‐graphs, whence recognizing hamiltonian L1‐graphs is an NP‐complete problem. The following results about L1‐graphs, unifying known results of Ore‐type and known results on K1,3‐free graphs, are obtained. Set K = lcub;G|Kp,p+1 (contained within) G (contained within) Kp V Kp+1 for some ρ ≥ } (v denotes join). If G is a 2‐connected L1‐graph, then G is 1‐tough unless G (element of) K. Furthermore, if G is as connected L1‐graph of order at least 3 such that |N(u) (intersection) N(v)| ≥ 2 for every pair of vertices u, v with d(u, v) = 2, then G is hamiltonian unless G ϵ K, and every pair of vertices x, y with d(x, y) ≥ 3 is connected by a Hamilton path. This result implies that of Asratian and Khachatrian. Finally, if G is a connected L1‐graph of even order, then G has a perfect matching. © 1996 John Wiley & Sons, Inc.

Local Conditions for Long Cycles in Graphs

“…Asratian and his colleagues [4][5][6][7] have obtained local analogues of Theorems 3.3-3.6. The main idea of their method is to use the structure of balls of small radii.…”

“…Then G is Hamiltonian. Theorem 3.9 was later generalized to the following, using the set of exceptions K from Theorem 3.5: Theorem 3.10 (Asratian, Broersma, van den Heuvel, and Veldman [5]). Let G be a connected graph on at least three vertices such that for every triple u, w, v with d(u, v) = 2 and w ∈ N (u) ∩ N (v) the following two properties hold:…”

Local Conditions for Cycles in Graphs

“…Then G is Hamiltonian. Theorem 3.9 was later generalized to the following, using the set of exceptions K from Theorem 3.5: Theorem 3.10 (Asratian-Broersma-van den Heuvel-Veldman [5]). Let G be a connected graph on at least three vertices such that for every triple u, w, v with d(u, v) = 2 and w ∈ N (u) ∩ N (v) the following two properties hold:…”

“…Asratian-Broersma-van den Heuvel-Veldman[5]). Let G be a connected graph on at least three vertices such that for every triple u, w, v with d(u, v) = 2 and w ∈ N (u) ∩ N (v) the following two properties hold:d(u) + d(v) ≥ |N (u) ∪ N (v) ∪ N (w)| − 1 and |N (u) ∩ N (v)| ≥ 2.…”

Cycles in graphs and related problems

“…Obviously, the members of this family are L 1 -graphs but not claw-free. Theorem 3.99 was further improved by Asratian et al [14].…”