Hamiltonian cycles and dominating cycles passing through a linear forest

Let GG be an (m+2)(m+2)-graph on nn vertices, and FF be a linear forest in GG with |E(F)|=m|E(F)|=m and ω1(F)=sω1(F)=s, where ω1(F)ω1(F) is the number of components of order one in FF. We denote by σ3(G)σ3(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ3σ3 conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ3(G)≥n+2m+2+max{s−3,0}σ3(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through FF is dominating. Using this result, we prove that if σ3(G)≥n+κ(G)+2m−1σ3(G)≥n+κ(G)+2m−1 then GG contains a hamiltonian cycle passing through FF. As a corollary, we obtain a result that if GG is a 3-connected graph and σ3(G)≥n+κ(G)+2σ3(G)≥n+κ(G)+2, then GG is hamiltonian-connected.