I think the answers already given by Thomas Kalinowski are more comprehensive, but Douglas B. West maintains another collection of open problems in graph theory at http://www.math.illinois.edu/~dwest/openp/ ([Wayback Machine](https://web.archive.org/web/20210507015418/https://faculty.math.illinois.edu/~west/openp/)) and there are also several graph theory problems listed at [List of unsolved problems in mathematics § Graph theory](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Graph_theory) on Wikipedia.