A note on hamiltonian cycles in 4-tough (P2 ∪ kP1)-free graphs

Hamiltonian cycles in 2‐tough 2K2 $2{K}_{2}$‐free graphs

“…We say that a graph G is (K 2 ∪ kK 1 )-free if it does not contain K 2 ∪ kK 1 as an induced subgraph. Recently, Shi and Shan conjectured that every 1-tough and 2k-connected (K 2 ∪ kK 1 )-free graph is hamiltonian [15]. In this paper, we solve this conjecture by proving the following stronger statement; every kconnected (K 2 ∪ kK 1 )-free graph of order n with minimum degree at least 3(k−1) 2 and independence number at most n 2 is hamiltonian or the Petersen graph.…”

“…In fact, for any constant m, the complete bipartite graph K m,m+1 is a non-hamiltonian m-connected (K 2 ∪ kK 1 )-free graph. However, in [15], Shi and Shan conjectured that the condition 4-tough of Theorem 1.4 is not sharp and can be relaxed to 1-tough.…”

Hamiltonicity of $1$-tough $(P_2\cup kP_1)$-free graphs

“…Bauer, Broersma and Veldman [2] showed that t 0 ≥ 9 4 if it exists. Conjecture 1 has been confirmed for a number of special classes of graphs [1,[3][4][5][6][9][10][11][12][13][14][15][16][17][18][19]. For example, it has been confirmed for graphs with forbidden (small) linear forests, such as 1-tough R-free graphs with R = P 3 ∪ P 1 , P 2 ∪ 2P 1 [13], 2-tough 2P 2 -free graphs [4,14,16], 3-tough (P 2 ∪3P 1 )-free graphs [11], 7-tough (P 3 ∪ 2P 1 )-free graphs [10] and 15-tough (P 3 ∪ P 2 )-free graphs [17].…”

“…Chvátal and Erdös [8] proved that, for any integer k ≥ 1, every max{2, k}-connected (k + 1)P 1 -free graph on at least three vertices is Hamiltonian. Along the Chvátal-Erdös theorem, Shi and Shan [18] proved that, for any integer k ≥ 4, every 4-tough 2kconnected (P 2 ∪kP 1 )-free graph with at least three vertices is Hamiltonian, and furthermore, they proposed a conjecture that for any integer k ≥ 1, any 1-tough 2k-connected (P 2 ∪kP 1 )-free graph is Hamiltonian. In this paper, we confirm the conjecture, and furthermore, we show that if k ≥ 3, then the condition '2k-connected' may be weakened to be '2(k − 1)-connected'.…”

An Ore-type condition for hamiltonicity in tough graphs and the extremal examples

“…Chvátal's toughness conjecture has been verified for certain classes of graphs including planar graphs, claw-free graphs, co-comparability graphs, and chordal graphs [2]. The classes also include 2K 2 -free graphs [6,14,12], (P 2 ∪ P 3 )-free graphs [15], and R-free graphs for R ∈ {P 2 ∪ P 3 , P 3 ∪ 2P 1 , P 2 ∪ kP 1 } [9, 15,16], where k ≥ 4 is an integer. In general, the conjecture is still wide open.…”