Abstract: We prove the Kauffman-Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p , every nontrivial Fox p -coloring of D will assign different colors to different arcs. 57M25 1 IntroductionIn 1999, Louis Kauffman and Frank Harary published a paper [5] detailing a graphtheoretic approach to the study of knot theory.In the paper they state a conjecture (Alternation Conjecture 6.2) that has come to be known as the Kauffman-Harary Conjecture:Conjecture 1 (Kauffman-Harary Conjecture) Let D be a reduced, alternating diagram of the knot k having prime determinant p .Then, every nontrivial p -coloring of D assigns different colors to different arcs.This provides a nice connection between two knot invariants, the determinant, which is relatively easy to calculate, for example, using the Alexander polynomial, and the least number of colors required on a minimal diagram of a knot, an invariant which is, in general, very difficult to evaluate.The conjecture asserts that in the case of an alternating knot, if the determinant is prime, then the least number of colors over minimal diagrams is simply the crossing number.Conjecture 1 is known to hold for rational knots by Kauffman and Lambropoulou [7] and Person et al [12], pretzel knots by Asaeda, Przytycki and Sikora [2], and many Turk's head knots by Dowdall et al [4].Our goal in this paper is to prove the conjecture for all alternating knots of prime determinant.In order to give an overview of our approach we recall some basic ideas about coloring.Let k be an alternating knot of prime determinant p .A (Fox) p -coloring of a diagram D of k is an assignment of integers mod p to the arcs of D such that the equation 2x y z Á 0 mod p holds at each crossing, x being the color of the over