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. Thomassen proposed a well-known conjecture: every 4-connected line graph is hamiltonian. In this note, we show that Thomassen’s conjecture is equivalent to the statement that the shortness coefficient of the class of all 4-connected line graphs is one and the statement that the shortness coefficient of the class of all 4-connected claw-free graphs is one respectively.
Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai (Combinatorics and Graph Theory, vol 95, World Scientific, Singapore, pp 53–69; Conjecture 8.6 of 1995) conjectured that every 3-edge connected and essentially 6-edge connected graph is collapsible. Denote D3(G) the set of vertices of degree 3 of graph G. For $${e = uv \in E(G)}$$ , define d(e) = d(u) + d(v)...
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