Global properties of graphs with local degree conditions

Abstract

Let P be a graph property. A graph G is said to be locally P (closed locally P, respectively) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property P. A graph G of order n is said to satisfy Dirac's condition if δ(G) ≥ n/2 and it satisfies Ore's condition if for every pair u, v of non-adjacent vertices in G, deg(u) + deg(v) ≥ n. A graph is locally Dirac (locally Ore, respectively) if the subgraph induced by the open neighbourhood of every vertex satisfies Dirac's condition (Ore's condition, respectively). In this paper we establish global properties for graphs that are locally Dirac and locally Ore. In particular we show that these graphs, of sufficiently large order, are 3-connected. For locally Dirac graphs it is shown that the edge connectivity equals the minimum degree and it is illustrated that this results does not extend to locally Ore graphs. We show that ⌊n/3⌋ − 1 is a sharp upper bound on the diameter of every locally Dirac graph of order n. We show that there exist infinite families of planar closed locally Dirac graphs. In contrast, locally Dirac graphs of sufficiently large order are shown to be non-planar. It is known that every closed locally Ore graph is hamiltonian. We show that locally Dirac graphs have an even richer cycle structure by showing that all locally Dirac graphs with maximum degree 11 are in fact fully cycle extendable. This result supports Ryjáček's well-known conjecture; which states that every connected, locally connected graph is weakly pancyclic.

Local connectivity, local degree conditions, some forbidden induced subgraphs, and cycle extendability

“…Furthermore, they observed that results of Hasratian and Khachatrian [7] imply that every connected locally Dirac graph of order at least 3 is hamiltonian. Our results generalize the mentioned results of Zhang [14] and Kubicka et al [9].…”

“…x∈N G (z)∩V (C) (|A k (x)| + |C k (x)|) ≤ x∈N G (z)∩V (C) |A(x) ∪ C(x)| ≤ x∈V (C) n(C) = n(C) 2 ,we obtain a contradiction for k > n(C) 2 , which completes the proof. ✷ Note that Corollary 10 extends the main result, Theorem 3.3, of Kubicka et al[9].Corollary 10 Every connected locally Ore graph G of order at least 3 is fully cycle extendable.Proof: Let G be as in the statement. Let vuw be an induced path of order 3 in G. By inclusion-exclusion, we obtaind G (u) = |{v, w}| + |(N G (v) ∪ N G (w)) ∩ N G (u)| + |N G (u) \ (N G [v] ∪ N G [w])| = 2 + |N G (u) ∩ N G (v)| + |N G (u) ∩ N G (w)| − |N G (u) ∩ N G (v) ∩ N G (w)| +|N G (u) \ (N G [v] ∪ N G [w])|.…”

“…Faudree et al [5] weakened the local connectivity requirement for this last result. Kubicka et al [9] considered locally Dirac graphs, and showed that every connected locally Dirac graph G with n(G) ≥ 3 and ∆(G) ≤ 11 is fully cycle extendable. Furthermore, they observed that results of Hasratian and Khachatrian [7] imply that every connected locally Dirac graph of order at least 3 is hamiltonian.…”