Mathematical Notes, vol. 72, no. 1, 2002, pp. 31–33. Translated from Matematicheskie Zametki, vol. 72, no. 1, 2002, pp. 35–37. Original Russian Text Copyright c 2002 by O. V. Borodin, A. V. Kostochka, A. Raspaud, E. Sopena. Estimating the Minimal Number of Colors in Acyclic k -Strong Colorings of Maps on Surfaces O. V. Borodin, A. V. Kostochka, A. Raspaud, and E. Sopena Received August 29, 2001 Abstract—A coloring of the vertices of a graph is called acyclic if the ends of each edge are colored in distinct colors, and there are no two-colored cycles. Suppose each face of rank k , k ≥ 4, in a map on a surface S N is replaced by the clique having the same number of vertices. It is proved in [1] that the resulting pseudograph admits an acyclic coloring with the number of colors depending linearly on N and k . In the present paper we prove a sharper estimate 55(−Nk) 4/7 for the number of colors provided that k ≥ 1 and −N ≥ 5 7 k 4/3 . Key words: graphs on surfaces, acyclic colorings, k-strong colorings. 1. INTRODUCTION A coloring of a graph embedded in a surface is called a k-strong coloring if any two vertices belonging to the boundary of the same face of rank not greater than k have different colors. A coloring of graph vertices is called acyclic if it is regular, i.e., the ends of each edge are colored in distinct colors, and there are no two-colored cycles. Acyclic colorings have a number of applications to other coloring problems (see [2–6]). A pseudograph embedded in a surface without faces of rank 1 and 2 is called a map. Note that a loop generates an edge with the same colors of the ends, and a double edge generates a two-colored cycle. We consider a coloring of vertices of a map on an arbitrary surface acyclic if the ends of each edge e which is not a loop have distinct colors and there are no two-colored cycles of length greater than 2. In [7] Alon, Mohar, and Sanders established the following estimate. Theorem 1. Every graph which admits an acyclic embedding in a surface S N is acyclically 50(2 − N ) 4/7 -colorable, and this estimate is sharp up to a factor of the form c(log(3 − N )) 1/7 . The proof of this statement was based on the following result due to Alon, MacDiarmid, and Reed [8]. Theorem 2 [8]. Each graph with degree of vertices not exceeding ∆ admits an acyclic coloring in 50∆ 4/3 colors. In [1], colorings which are both acyclic and k-cyclic were discussed. Namely, we studied an acyclic coloring of the pseudograph G (k) obtained by replacing each face of rank not greater than k of a map G with the clique having the same set of vertices. This means that each face of rank not greater than k acquires all “invisible diagonals.” For k = 3 this coloring coincides with the acyclic coloring. The main result of [1] is as follows. 0001-4346/2002/7212-0031$27.00 c 2002 Plenum Publishing Corporation 31