### Some problems related to hamiltonian line graphs [chapter]

2007 Proceedings of the International Conference on Complex Geometry and Related Fields
Part of this paper summarizes some of the recent developments in the study of hamiltonian line graphs and the related hamiltonian claw-free graphs. The last section of this paper solves some problems on the hamiltonian like indices from a paper by Clark and Wormald in 1983. Definitions and Terminology Graphs considered here are finite and loopless. Unless otherwise noted, we follow  for notations and terms. As in , κ(G), κ (G) and δ(G) represent the connectivity, edge-connectivity, and
more » ... minimum degree of a graph G, respectively. Definition 1.2. A vertex cut X of G is essential if G − X has at least two nontrivial components. Definition 1.3. For an integer k > 0, a graph G is essentially k-connected if G does not have an essential cut X with |X| < k. Definition 1.4. An edge cut Y of G is essential if G − Y has at least two nontrivial components. Definition 1.5. For an integer k > 0, a graph G is essentially k-edge-connected if G does not have an essential edge cut Y with |Y | < k. Definition 1.6. Let G be a graph and let X ⊆ E(G) be an edge subset. The contraction G/X is the graph obtained from G by identifying the two ends of each edge in X and then deleting the resulting loops.