Degree sums and subpancyclicity in line graphs

A graph is called subpancyclic if it contains a cycle of length k for each k between 3 and the circumference of the graph. In this paper, we show that if the degree sum of the vertices along each 2-path of a graph G exceeds (n+6)/2, or if the degree sum of the vertices along each 3-path of G exceeds (2n+16)/3, then its line graph L(G) is subpancyclic. Simple examples show that these bounds are best possible. Our results shed some light on the content of a famous Metaconjecture of Bondy

Circuits in graphs and the Hamiltonian index

This thesis contains some results on circuits and the hamiltonian index of a graph. In the first part of the thesis, Chapter 2 through 4, we concentrate on the topic of the hamiltonian index of a graph. In the second part, Chapters 5 through 9, we focus on the topic of the degree sum along paths for subpancyclic line graphs. The results in these chapters are all related to line graphs and are best possible. In Chapter 10, we obtain some results on so-called connected even factors with degree restriction. In the last chapter, we obtain some degree conditions for supereulerian graphs. These results are also best possible