Collapsible graphs and Hamilton cycles of line graphs

For a graph GG, let O(G)O(G) denote the set of odd degree vertices of GG. A graph GG is collapsible   if for any subset R⊆V(G)R⊆V(G) with |R|≡0|R|≡0 (mod 2), GG has a spanning connected subgraph HRHR such that O(HR)=RO(HR)=R. The reduction of GG is the graph obtained from GG by contracting all maximally collapsible subgraphs until no collapsible subgraph is left. Let GG be a graph on n≥8n≥8 vertices. In this paper, we prove that if d(x)+d(y)≥n−2−p(n)d(x)+d(y)≥n−2−p(n) for each xy∈E(G)xy∈E(G), then GG is collapsible or GG is one of 43 special graphs or the reduction of GG is K1,tK1,t where t≥2t≥2 or GG is a class of well-characterized graphs, where p(n)=0p(n)=0 for nn even and p(n)=1p(n)=1 for nn odd, which generalizes the earlier results by Catlin (1987), and by Li and Yang (2012).