An Extremal Property of Tur´ an Graphs, II Spencer N. Tofts DEPARTMENT OF MATHEMATICS UNIVERSITY OF PENNSYLVANIA PHILADELPHIA, PA 19104 E-mail: spencert@math.upenn.edu Received March 14, 2012; Revised January 27, 2013 Published online 1 May 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jgt.21739 Abstract: Let T 2 (n) denote Tur´ an’s graph—the complete 2-partite graph on n vertices with partition sizes as equal as possible. We show that for all n ≥ 4, the graph T 2 (n) has more proper vertex colorings in at most 4 colors than any other graph with the same number of vertices and edges. C  2013 Wiley Periodicals, Inc. J. Graph Theory 75: 275–283, 2014 Keywords: Tur´ an graph; extremal graph; number of colorings; chromatic polynomial Mathematics Subject Classification: 05C15; 05C30; 05C31; 05C35 1. INTRODUCTION All graphs in this article are finite, undirected, and have neither loops nor multiple edges. For all missing definitions and basic facts which are mentioned but not proved, we refer the reader to Bollob´ as [3]. For a graph G, let V = V (G) and E = E (G) denote the vertex set of G and the edge set of G, respectively. Let |A| denote the cardinality of a set A. Let n = n(G) =| V (G)| and e = e(G) =|E (G)| denote the number of vertices (the order) of G, and number of edges (the size) of G, respectively. An edge {x, y} of G will also be denoted by xy, or yx. For sets Dedicated to the memory of Professor Herbert S. Wilf. Journal of Graph Theory C  2013 Wiley Periodicals, Inc. 275