where "u" and "v" are adjacent vertices, and formula_69 is the graph with the edge removed. formula_70 represents the number of possible proper colorings of the graph, where the vertices may have the same or different colors. Then the proper colorings arise from two different graphs. To explain, if the vertices "u" and "v" have different colors, then we might as well consider a graph where "u" and "v" are adjacent. If "u" and "v" have the same colors, we might as well consider a graph where "u" and "v" are contracted. Tutte’s curiosity about which other graph properties satisfied this recurrence led him to discover a bivariate generalization of the chromatic polynomial, the Tutte polynomial.