Unavoidable set of face types for planar maps
Discussiones Mathematicae Graph Theory
The type of a face f of a planar map is a sequence of degrees of vertices of f as they are encountered when traversing the boundary of f . A set T of face types is found such that in any normal planar map there is a face with type from T . The set T has four infinite series of types as, in a certain sense, the minimum possible number. An analogous result is applied to obtain new upper bounds for the cyclic chromatic number of 3-connected planar maps. , Zaks  . Recently, unifying and
... ngthening Kotzig's results  , Borodin  has proved that any planar triangulation without vertices of degree 4 contains either a triangle of weight (the degree sum of its incident vertices) at most 29 incident with a 3-valent vertex or a triangle whose weight does not exceed 17. A sharp inequality for the number of triangles of weight at most 17 in planar maps with minimum degree 5 was found by Borodin  . Edges of small weights in planar maps of minimum degree 5 are investigated in a very recent paper by Borodin and Sanders . Both the above mentioned papers complete the work contributed to by many authors, among others Grünbaum , Kotzig , Fisk (see ), Wernicke . Many structural results on planar maps have been obtained by solving some colouring problems, see e.g. Borodin [1, 3, 6, 7] , Jendrol' and Skupień  . The main aim of this paper is to prove an analogue of Lebesgue's theorem, which is optimal in a certain sense. Fundamentals For integers p, q we denote by [p, q] the set of all integers i, p ≤ i ≤ q, and by [p, ∞) the set of all integers ≥ p. A finite sequence Q is said to be equivalent to a finite sequence P if Q can be obtained from P using rotation and/or mirror image. Thus, if P = (p 1 , . . . , p n ), then Q = (p 1+i , . . . , p n+i ) or Q = (p n−i , . . . , p 1−i ) for some i ∈ [0, n − 1], where indices are taken modulo n. (We use this "modulo convention" throughout the whole paper.) Let P, P 1 , P 2 be finite sequences and let m ∈ [1, ∞). We denote by P 1 P 2 the concatenation of P 1 and P 2 (in that order), by P m the m-fold concatenation of P 's and by len(P ) the length of P .