On minimum degree, leaf number, traceability and Hamiltonicity in graphs

Connectivity, Traceability and Hamiltonicity

“…Sufficient conditions for a graph to be Hamiltonian has a long history [4,5,18,19]. The search for sufficient conditions for a graph to be Hamiltonian remains an on going research area (see for instance [12,16,17]). It is known that for a graph to be Hamiltonian necessarily, it must be 2-connected.…”

“…Theorem 1 was shown to be best in the sense that, for each δ, the graph G 2δ+1 = K δ + K 1 + K δ contains a cut-vertex and has leaf number L(G 2δ+1 ) = 2δ. For graphs in general, Theorem 1 has been used as a powerful tool in the establishment of sufficient conditions for traceability or Hamiltonicity based on leaf number and minimum degree (see for instance [11][12][13][14]16]). This motivated us to establish a similar bound for triangle-free graphs with the hope that the results will in the near future play an important role in providing sufficient conditions for triangle-free graphs to be either Hamiltonian or traceable.…”

Estimating the circumference of a graph in terms of its leaf number

“…After that, Mukwembi [15] proved that if G is a connected claw-free graph with δ(G) ≥ (L(G) + 1)/2, then G is hamiltonian. In recent years, several authors reported on sufficient conditions for a graph to be hamiltonian or traceable based on minimum degree and leaf number, see [9][10][11][12][13].…”