Collapsible Graphs and Hamiltonicity of Line Graphs
Résumé
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 $e=uv∈E(G)$, define $d(e)=d(u)+d(v)−2$ the edge degree of $e$, and $\xi(G)=\min\{d(e):e∈E(G) \}$. Denote by $\lambda^m(G)$ the $m$-restricted edge-connectivity of $G$. In this paper, we prove that a 3-edge-connected graph with $\xi(G)\geq 7$, and $\lambda^3(G)\geq 7$ is collapsible; a 3-edge-connected simple graph with $\xi(G)\geq 7$, and $\lambda^3(G)\geq 6$ is collapsible; a 3-edge-connected graph with $\xi(G)\geq 6, \lambda^2(G)\geq 4$, and $\lambda^3(G)\geq 6$ with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected simple graph with $\xi(G)\geq 6$, and $\lambda^3(G)\geq 5$ with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected graph with $\xi(G)\geq 5$ and $\lambda^2(G)\geq 4$ 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 $|D_3(G)|\leq 9$ is Hamiltonian.