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/ and there are also several graph theory problems listed at https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Graph_theory

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) and there are also several graph theory problems listed at List of unsolved problems in mathematics § Graph theory on Wikipedia.