text: This quantity's associated polynomial is often calculated using the deletion-contraction algorithm.
full_question: This quantity's associated polynomial is often calculated using the deletion-contraction algorithm. Heinrich Tietze was the first to determine this quantity for the projective plane, since he found that the graph "K sub 6" was the largest graph that could be embedded onto it. If a graph has a value of n for this quantity, then it will contract to the complete graph of n, possibly with some extra adjacency edges, according to Hadwiger's conjecture. For a surface, this quantity is less than "seven plus the square root of 49 minus 24 times the Euler characteristic, all over 2." Removing the critical edge of a graph will decrease this quantity. Kenneth Appel and Wolfgang Haken proved that this quantity is equal to four for planar graphs. For 10 points, name this quantity, which is the minimum number of colors needed to color a graph so that no two adjacent vertices have the same color.
first_sentence: This quantity's associated polynomial is often calculated using the deletion-contraction algorithm.
raw_answer: chromatic number [prompt on "the number of colors needed to color a graph"]