Cyclic chromatic number of 3-connected plane graphs

Let G be a 3-connected plane graph. Plummer and Toft [J. Graph Theory, 11 ( 1987), pp. 507-515] conjectured that chi (c)(G) less than or equal to Delta *(G) + 2, where chi (c)(G) is the cyclic chromatic number of G and Delta*(G ) the maximum face size of G. Hornak and Jendrol' [J. Graph Theory, 30 (199 9), pp. 177-189] and Borodin and Woodall [SIAM J. Discrete Math., submitted ] independently proved this conjecture when ( G) is large enough. Moreover, Borodin and Woodall proved a stronger statement that chi (c)(G) less than or equal to Delta*(G) + 1 holds if Delta*(G) greater than or equal to 122. In this paper, we prove that chi (c)(G) less than or equal to Delta*(G) + 1 holds if Delta*(G) greater than or equal to 60.