Abstract: Graph theory is now an established discipline but the study of graph homomorphisms has only recently begun to gain wide acceptance and interest. This text is devoted entirely to the subject, bringing together the highlights of the theory and its many applications. It looks at areas such as graph reconstruction, products, fractional and circular colourings, and constraint satisfaction problems, and has applications in complexity theory, artificial intelligence, telecommunications, and statistical physics. It has a wide focus on algebraic, combinatorial, and algorithmic aspects of graph homomorphisms. A reference list and historical summaries extend the material explicitly discussed. The book contains exercises of varying difficulty. Hints or references are provided for the more difficult exercises.

Keywords: digraphs, partial orders, reconstruction, colourings, fractional colourings, products, algebraic combinatorics, structures, complexity theory, constraint satisfaction