Reviews

Abstract

The most revelatory experience of my undergraduate mathematics degree was a writing course offered by James Vick. Well on his way to becoming a vice president in the University of Texas administration, Vick still managed to teach the occasional class. This one, “Proofs, Conjectures, and Controversies,” showed us that mathematics was not always as absolute as implied in our other courses. In particular, there is not always consensus about what constitutes a valid proof and what tools are allowed. One example of controversy was the 1976 computer-assisted proof of the four color theorem, establishing that the countries of every map can be colored with at most four colors so that no two neighboring countries are colored the same. The class readings included the recent Dover edition of Saaty and Kainan’s book [18] on the “assaults and conquest” of that graph theory problem and Leonard Gillman’s celebrated guide to mathematical writing [5]. What an honor to learn about recent and contentious research developments while developing skills in mathematical exposition. The year 1976 was an important year for both the United States and graph theory. In that year of the American bicentennial, Kenneth Appel and Wolfgang Haken of the University of Illinois solved Francis Guthrie’s 1852 four color conjecture. This was one of the rare mathematical results to be mentioned (eventually) in The New York Times [19, p. 209]. Another milestone for graph theory that year was the publication of the first book on its history, Graph Theory 1736–1936 by Norman Biggs, Keith Lloyd, and Robin Wilson [2]. That book opened a sort of trilogy on the history of graph theory, all involving the talented and prolific expositor Robin Wilson. The second book was a 2002 solo effort Four Colors Suffice [19]. The third book, based on the 2012 dissertation of David Parks [16] and also featuring John Watkins, focuses on the period 1876–1976 with an emphasis on US and Canadian contributions to the field. This review will focus on the 2023 book while also discussing it in relation to its two predecessors.