Dirac's minimum degree condition restricted to claws

Dirac's theorem for random graphs

A classical theorem of Dirac from 1952 asserts that every graph on n vertices with minimum degree at least dn/2e is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of random graph properties we prove that if p log n/n, then a.a.s. every subgraph ofG(n, p) with minimum degree at least (1/2+o(1))np is Hamiltonian. Our result improves on prev...

Vertex-disjoint t-claws in graphs

Let δ(G) denote the minimum degree of a graph G. We prove that for t ≥ 4 and k ≥ 2, a graph G of order at least (t + 1)k + 11 6 t 2 with δ(G) ≥ k + t − 1 contains k pairwise vertex-disjoint copies of K1,t.

A Degree Condition For Graphs to Have Connected (g, f)-Factors

Sufficient conditions for a graph to be super restricted edge-connected

Restricted edge connectivity is a more refined network reliability index than edge connectivity. A restricted edge cut F of a connected graph G is an edge cut such that G−F has no isolated vertex. The restricted edge connectivity λ′ is theminimumcardinality over all restricted edge cuts.WecallG λ′-optimal if λ′ = ξ , where ξ is theminimum edgedegree inG. Moreover, a λ′-optimal graphG is called ...

Forbidden Pairs for Vertex-Disjoint Claws

Let k ≥ 4, and let H1, H2 be connected graphs with |V (Hi)| ≥ 3 for i = 1, 2. A graph G is said to be {H1, . . . , Hl}-free if none of H1, . . . , Hl is an induced subgraph of G. We prove that if there exists a positive integer n0 such that every {H1, H2}-free graph G with |V (G)| ≥ n0 and δ(G) ≥ 3 contains k vertex-disjoint claws, then {H1, H2} ∩ {K1,t | t ≥ 2} = ∅. Also, we prove that every K...