This thesis deals with graph theory, a branch of mathematics that studies the structure and characteristics of graphs, as well as its application in additional mathematics education. The introductory section presents basic concepts and definitions necessary for understanding the thesis. The second chapter focuses on the problem of the seven bridges of the city of Konigsberg, which was solved by Leonhard Euler in the first half of the 18th century, thus laying the foundation for graph theory. Here, definitions and theorems related to Eulerian and semi-Eulerian graphs are also given. In the third chapter, the handshaking problem is discussed in more detail. The fourth chapter is based on the problem of map coloring, that is, whether we can color a (geographical) map with a maximum of four colors so that neighboring countries are colored differently. The thesis emphasizes the importance of studying graph theory in mathematics and how to make it more accessible to students of various ages. Graph theory does not belong to the regular mathematics curriculum, so the suggested activities in the thesis can be used in additional teaching or as part of mathematical projects.