Discrete Mathematics 50 (1984) 231-237 North-Holland 231 ON A PROBLEM OF J. ZAKS CONCERNING 5-VALENT 3-CONNECTED PLANAR GRAPHS Stanislav JENDROL, Department of Mathematics, P.J. Safdrik University, 041 54 Ko~ice, Czechoslovakia Received 29 July 1983 Recently J. Zaks formulated the following Eberhard-type problem: Let (Ps, P6 .... ) be a finite sequence of nonnegative integers; does there exist a 5-valent 3-connected planar graph G such that it has exactly Pk k-gons for all k ~> 5, m i of its vertices meet exactly i triangles, 4 ~< i <~ 5, and m4+2ms=24+3 ~ (k-4)pk ? This paper brings a solution to the problem, and similar problems are considered as well. 1. Introduction Let ~3(s, g) be the set of s-connected graphs embedded in an orientable surface Sg of genus g which contain no loops or multiple edges, s~ > 2, g I>0. For G~ ~(s, g) let pk(G) or vk(G) denote the number of k-gonal faces or k-valent vertices of G respectively, where k I>3. Further let M~(G) denote the set of those vertices of G which are incident with exactly i triangles, i >~0; let m~(G)= IMp(G)[. Let G(r, s, g) be the subset of ~d(s, g) considering of regular graphs of degree r. Evidently by Euler's relation, (2r+2k-rk)pk = 4r(1- g). (E,) k~3 Kotzig [12] showed that Ma(G ) LI Ms(G ) ~ ~ for every graph G from ~(5, 3, 0). Zaks [17] proved the following extension of Kotzig's theorem: Theorem 1. If G ~ c~(5, 3, g), g I>0, then m4+2ms>~24(1-g)+3 ~, (k-4)pk; equality holds if and only if mo= ml = mz = O. (1) In the same paper the following Eberhard-typ¢ problem is formulated: Let (Ps, P6, • -.) be a finite sequence of nonnegative integers; does there exist a graph G ~ ~(5, 3, 0) such that I~(G) = 1~ for all k >~ 5 and m4(G)+ 2ms(G)= 24+ 3 ~., (k-4)pk ? k~5 0012-365X/84/$3.00 ~) 1984, Elsevier Science Publishers B.V. (North-Holland)