Traceability on 2-connected line graphs

Abstract In this paper, we mainly prove the following: Let G be a connected almost bridgeless simple graph of order n sufficiently large such that σ ¯ 2 ( G ) =min { d ( u ) + d ( v ) : u v ∈ E ( G ) } ≥ 2 ( ⌊ n / 11 ⌋ − 1 ) . Then either L(G) is traceable or Catlin’s reduction of the core of G is one of eight graphs of order 10 or 11, where the core of G is obtained from G by deleting the vertices of degree 1 of G and replacing each path of length 2 whose internal vertex has degree 2 in G by an edge. We also give a new proof for the similar theorem in Niu et al. (2012) which has flaws in their proof.