Some problems related to hamiltonian line graphs

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. 1. Definitions and Terminology Graphs considered here are finite and loopless. Unless otherwise noted, we follow [2] for notations and terms. As in [2], κ(G), κ′(G) and δ(G) represent the connectivity, edge-connectivity, and the minimum degree of a graph G, respectively. Definition 1.1. A graph G is nontrivial if E(G) 6= ∅. 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. For convenience, we use G/e for G/{e} and G/∅ = G; and if H is a subgraph of G, we write G/H for G/E(H). Definition 1.7. For a graph G, O(G) denotes the set of all vertices of odd degree in G. Definition 1.8. A graph G is even if O(G) = ∅, is eulerian if G is both even and connected, and is supereulerian if G contains a spanning eulerian subgraph. 1991 Mathematics Subject Classification. Primary 05C45; Secondary 05C38.