Collapsible Graphs and Hamiltonicity of Line Graphs

Abstract

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 D ₃(G) the set of vertices of degree 3 of graph G. For ${e = uv \in E(G)}$, define d(e) = d(u) + d(v) − 2 the edge degree of e, and ${\xi(G) = \min\{d(e) : e \in E(G)\}}$. Denote by λ^m(G) the m-restricted edge-connectivity of G. In this paper, we prove that a 3-edge-connected graph with ${\xi(G)\geq7}$, and ${\lambda^3(G)\geq7}$ is collapsible; a 3-edge-connected simple graph with ${\xi(G)\geq7}$, and ${\lambda^3(G)\geq6}$ is collapsible; a 3-edge-connected graph with ${\xi(G)\geq6}$, ${\lambda^2(G)\geq4}$, and ${\lambda^3(G)\geq6}$ with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected simple graph with ${\xi(G)\geq6}$, and ${\lambda^3(G)\geq5}$ with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected graph with ${\xi(G)\geq5}$, and ${\lambda^2(G)\geq4}$ with at most 9 vertices of degree 3 is collapsible. As a corollary, we show that a 4-connected line graph L(G) with minimum degree at least 5 and ${|D_3(G)|\leq9}$ is Hamiltonian.