We study the distributed message-passing model in which a communication network is represented by a graph G=(V,E). Usually, the measure of complexity that is considered in this model is the worst-case complexity, which is the largest number of rounds performed by a vertex v\in V. While often this is a reasonable measure, in some occasions it does not express sufficiently well the actual performance of the algorithm. For example, an execution in which one processor performs r rounds, and all the rest perform significantly less rounds than r, has the same running time as an execution in which all processors perform the same number of rounds r. On the other hand, the latter execution is less efficient in several respects, such as energy efficiency, task execution efficiency, local-neighborhood efficiency and simulation efficiency. Consequently, a more appropriate measure is required in these cases. Recently, the vertex-averaged complexity was proposed by \cite{Feuilloley2017}. In this measure, the running time is the worst-case sum of rounds of communication performed by all of the graph's vertices, averaged over the number of vertices. Feuilloley \cite{Feuilloley2017} showed that leader-election admits an algorithm with a vertex-averaged complexity significantly better than its worst-case complexity. On the other hand, for O(1)-coloring of rings, the worst-case and vertex-averaged complexities are the same. This complexity is O\left(\log^{*}n\right) [12]. It remained open whether the vertex-averaged complexity of symmetry-breaking in general graphs can be better than the worst-case complexity. We answer this question in the affirmative, by showing a number of results with improved vertex-averaged complexity.