3-Connected {K 1,3, P 9}-Free Graphs are Hamiltonian-Connected

Hamiltonian extendable graphs

“…, 3, 5) is not Hamiltonian extendable, contradicting the assumption that G is Hamiltonian extendable. This proves (2).…”

“…D 1 = ∅ and D 2 = ∅. Note that the case when A 1 = ∅, B 1 = ∅ and the case when A 1 = ∅, B 1 = ∅ are symmetric; by Claim 15 (2), the case when A 1 = ∅, B 1 = ∅ and the case when A 1 = ∅, B 1 = ∅ are symmetric.…”

“…Then C 1 = ∅ because G is 2-connected. By Claim 15 (2) and (3), B 2 = ∅, A 2 = ∅. Since {u, v} is a minimum vertex cut of G, we have C 2 = ∅.…”

“…However, the converse is not true in general. In [2,4,9,15], the authors have partially characterized pairs of forbidden subgraphs for a 3-connected graph to be Hamiltonian connected. Note that the 3-connected condition is necessary for Hamiltonian connected graphs while not necessary for Hamiltonian extendable graphs.…”

The maximum degree of a minimally hamiltonian-connected graph

“…Clearly every hamiltonian-connected graph contains a minimally hamiltonian-connected spanning subgraph. Concerning the maximum degree of a minimally hamiltonian-connected graph with a given order, Modalleliyan and Omoomi [2] proved the following results: (1) The maximum degree of any minimally hamiltonian-connected graph of order n is not equal to n − 2; (2) the wheel W n is the only minimally hamiltonian-connected graph of order n with maximum degree n − 1; (3) for every integer n ≥ 6 and any integer ∆ with n/2 ≤ ∆ ≤ n − 3, there exists a minimally hamiltonian-connected graph of order n with maximum degree ∆. They [2] posed the question of whether for ∆ in the range 3 ≤ ∆ < n/2 , there exists a minimally hamiltonian-connected graph of order n with maximum degree ∆.…”