Long cycles in graphs through fragments

Four basic Dirac-type sufficient conditions for a graph G to be hamiltonian are known involving order n, minimum degree δ, connectivity κ and independence number α of G: (1) δ ≥ n/2 (Dirac); (2) κ ≥ 2 and δ ≥ (n+κ)/3 (by the author); (3) κ ≥ 2 and δ ≥ max{(n+2)/3, α} (NashWilliams); (4) κ ≥ 3 and δ ≥ max{(n+2κ)/4, α} (by the author). In this paper we prove the reverse version of (4) concerning the circumference c of G and completing the list of reverse versions of (1)-(4): (R1) if κ ≥ 2, then c ≥ min{n, 2δ} (Dirac); (R2) if κ ≥ 3, then c ≥ min{n, 3δ − κ} (by the author); (R3) if κ ≥ 3 and δ ≥ α, then c ≥ min{n, 3δ − 3} (Voss and Zuluaga); (R4) if κ ≥ 4 and δ ≥ α, then c ≥ min{n, 4δ − 2κ}. To prove (R4), we present four more general results centered around a lower bound c ≥ 4δ − 2κ under four alternative conditions in terms of fragments. A subset X of V (G) is called a fragment of G if N(X) is a minimum cut-set and V (G)− (X ∪N(X)) 6= ∅.