Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an ( m + 2 ) -graph on n vertices, and F be a linear forest in G with | E ( F ) | = m and ω 1 ( F ) = s , where ω 1 ( F ) is the number of components of order one in F . We denote by σ 3 ( G ) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ 3 conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ 3 ( G ) ≥ n + 2 m + 2 + max { s − 3 , 0 } , then every longest cycle passing through F is dominating. Using this result, we prove that if σ 3 ( G ) ≥ n + κ ( G ) + 2 m − 1 then G contains a hamiltonian cycle passing through F . As a corollary, we obtain a result that if G is a 3-connected graph and σ 3 ( G ) ≥ n + κ ( G ) + 2 , then G is hamiltonian-connected.