Several results concerning existence of k-paths, for which the sum

. We prove that each polyhedral triangular face free map G on a compact 2dimensional manifold M with Euler characteristic Ø(M) contains a k-path, i.e. a path on k vertices, such that each vertex of this path has, in G, degree at most 5 2 k if M is a sphere S 0 and at most k 2 ¯ 5+ p 49\Gamma24Ø(M 2 if M 6= S 0 or does not contain any k-path. We show that for even k this bound is best possible. Moreover, we verify that for any graph other than a path no similar estimation exists. 1. Introduction Throughout this paper we shall consider connected graphs without loops or multiple edges. Let P r denote a path on r vertices (an r-path in the sequel). For graphs H and G, G ¸ = H denotes that the graphs H and G are isomorphic. The standard notation \Delta(G) stands for the maximum degree of a graph G. For a vertex X of a graph G deg G (X) denotes the degree of X in G. Let H be a family of graphs and let H be a graph which is isomorphic to a subgraph of at least one member of H...

INTRODUCTION The k-dimensional skeleton of a d-polytope P is the set of all faces of the polytope of dimension at most k. The 1-skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs--- subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t

Abstract: The M-degree of an edge xy in a graph is the maximum of the degrees of x and y. TheM-degree of a graph G is the minimum over M-degrees of its edges. In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4-cycles has M-degree at most 8 and gave an example of such a graph with M-degree 3. This yields upper bounds on the game chromatic number of C4-free planar graphs. We determine the maximum possible M-degrees for

It is proved that by deleting at most 5 edges every planar graph can be reduced to a graph having a non-trivial automorphism. Moreover, the bound of 5 edges cannot be lowered to 4. 1 Introduction A graph G is called asymmetric if it admits no non-trivial automorphism. Asymmetry is the typical behaviour of finite graphs. In 1963, Erdos and R'enyi [2] proved that almost all graphs are asymmetric. They further proved in [2] that if s(n) is the maximum number of edges which must be added to and/or deleted from, a graph with n vertices in order The work of this author was supported by grant 97-01-01075 from the Russian Foundation of Fundamental Research. y The work of this author was supported by the program "Universities of Russia --- Fundamental Research" (project code 1792). z The work of this author was supported by grant 99-01-00581 from the Russian Foundation of Fundamental Research. x The work of this author was supported by grant OPG--7315 from the Natural Sciences and Eng...

Let # average degree, and # minimum degree of agraph An edge is light ifboth its endpoints hnd degree bounded by a constant depending only on # and #.

We define k-diverse colouring of a graph to be a proper vertex colouring in which every vertex x, sees min{k, d(x)} different colours in its neighbors. We show that for given k there is an f(k) for which every planar graph admits a k-diverse colouring using at most f(k) colours. Then using this colouring we obtain a K5-free graph H for which every planar graph admits a homomorphism to it, thus another proof for the result of J. Neˇsetˇril, P. Ossona de Mendez. 1

The gravity g(H; H) of a graph H in the family of graphs H is the greatest integer n with the property that for every integer m, there are in nitely many graphs G 2 H such that each subgraph of G, which is isomorphic to H, contains at least n vertices of degree m in G. We study the basic properties of the gravity function for various families of plane graphs. We also introduce and study the almost-light graphs and the absolutely heavy graphs. The paper concludes with few open problems.

Let G be a K 1;r -free graph (r 3) on n vertices. We prove that, for any induced path or induced cycle on k vertices in G (k 2r \Gamma 1 or k 2r, respectively), the degree sum of its vertices is at most (2r \Gamma 2)(n \Gamma ff) where ff is the independence number of G. As a corollary we obtain an upper bound on the length of a longest induced path and a longest induced cycle in a K 1;r -free graph. Stronger bounds are given in the special case of claw-free graphs (i.e. r = 3). Sharpness examples are also presented. c fl ??? John Wiley & Sons, Inc. 1. INTRODUCTION Claw-free graphs have been a subject of interest of many authors in the last years (see e.g. a recent survey by R. Faudree et al [6]). For this class of graphs we investigate problems which have their origin in the theory of planar graphs. Throughout the paper we use the most common graph theoretical terminology. For the concepts not defined here we refer to [1]. A graph G is called K 1;r -free if there is no induced su...

Let G be a K 1;r -free graph (r 3) on n vertices. We prove that, for any induced path or induced cycle on k vertices in G (k 2r \Gamma 1 or k 2r, respectively), the degree sum of its vertices is at most (2r \Gamma 2)(n \Gamma ff) where ff is the independence number of G. As a corollary we obtain an upper bound on the length of a longest induced path and a longest induced cycle in a K 1;r -free graph. Stronger bounds are given in the special case of claw-free graphs (i.e. r = 3). Sharpness examples are also presented. c fl ??? John Wiley & Sons, Inc. 1. INTRODUCTION Claw-free graphs have been a subject of interest of many authors in the last years (see e.g. a recent survey by Faudree et al. [6]). For this class of graphs we investigate problems which have their origin in the theory of planar graphs. Throughout the paper we use the most common graph theoretical terminology. For the concepts not defined here we refer to [1]. A graph G is called K 1;r -free if there is no induced sub...