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.