Here is a solid introduction to graph theory, covering Dirac's theorem on k-connected graphs, Harary-Nashwilliam's theorem on the hamiltonicity of line graphs, Toida-McKee's characterization of Eulerian graphs, Fournier's proof of Kuratowski's theorem on planar graphs, and more. The book does not presuppose deep knowledge of any branch of mathematics, but requires only the basics of mathematics.
PRELIMINARY TEXT: DO NOT USE Graph theory has experienced a tremendous growth during the course of this century. One of the main reasons for this phenomenon is the applicability of graph theory in other disciplines such as physics, chemistry, psychology, socioloy, and theoretical computer science. This book aims to provide a solid background in the basic topics of graph theory. It does not presuppose knowledge of any branch of mathematics, except some exposure to the theory of sets, and can be used in an advanced undergraduate or beginning graduate course in graph theory.
R. Balakrishnan