Edge-Face Chromatic Number and Edge Chromatic Number of Simple Plane Graphs Rong Luo 1 and Cun-Quan Zhang 2 1 DEPARTMENT OF MATHEMATICAL SCIENCES MIDDLE TENNESSEE STATE UNIVERSITY MURFREESBORO, TENNESSEE, 37132 2 DEPARTMENT OF MATHEMATICS WEST VIRGINIA UNIVERSITY MORGANTOWN, WEST VERGINIA, 26506-6310 E-mail: rluo@mtsu.edu, cqzhang@math.wvu.edu Received July 12, 2002; Revised August 26, 2004 Published online 4 April 2005 in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/jgt.20077 Abstract: Given a simple plane graph G, an edge-face k -coloring of G is a function  : E (G) [ F (G) !f1, ... ,k g such that, for any two adjacent or incident elements a, b 2 E (G) [ F (G), (a) 6¼ (b). Let  e (G),  ef (G), and Á(G) denote the edge chromatic number, the edge-face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that  ef (G) ¼  e (G) ¼ Á(G) for any 2-connected simple plane graph G with Á(G)  24. ß 2005 Wiley Periodicals, Inc. J Graph Theory 49: 234–256, 2005 Keywords: edge chromatic number; edge-face chromatic number; edge-face coloring —————————————————— Contract grant sponsor: National Security Agency; Contract grant number: MDA904-00-1-00614 (to C.-Q.Z.). ß 2005 Wiley Periodicals, Inc. 234