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Light Paths in 4Connected Graphs in the Plane and Other Surfaces
 J. Graph Theory
, 1998
"... Several results concerning existence of kpaths, for which the sum ..."
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Several results concerning existence of kpaths, for which the sum
On 3Connected Plane Graphs Without Triangular Faces
, 1998
"... . We prove that each polyhedral triangular face free map G on a compact 2dimensional manifold M with Euler characteristic Ø(M) contains a kpath, 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\Gam ..."
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. We prove that each polyhedral triangular face free map G on a compact 2dimensional manifold M with Euler characteristic Ø(M) contains a kpath, 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 kpath. 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 rpath 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...
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton 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 i ..."
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton 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
Mdegrees of quadranglefree planar graphs
 J. Graph Theory
"... Abstract: The Mdegree of an edge xy in a graph is the maximum of the degrees of x and y. TheMdegree of a graph G is the minimum over Mdegrees 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 4cycles has Mdegree ..."
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Abstract: The Mdegree of an edge xy in a graph is the maximum of the degrees of x and y. TheMdegree of a graph G is the minimum over Mdegrees 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 4cycles has Mdegree at most 8 and gave an example of such a graph with Mdegree 3. This yields upper bounds on the game chromatic number of C4free planar graphs. We determine the maximum possible Mdegrees for
Deeply Asymmetric Planar Graphs
, 2000
"... It is proved that by deleting at most 5 edges every planar graph can be reduced to a graph having a nontrivial automorphism. Moreover, the bound of 5 edges cannot be lowered to 4. 1 Introduction A graph G is called asymmetric if it admits no nontrivial automorphism. Asymmetry is the typical be ..."
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It is proved that by deleting at most 5 edges every planar graph can be reduced to a graph having a nontrivial automorphism. Moreover, the bound of 5 edges cannot be lowered to 4. 1 Introduction A graph G is called asymmetric if it admits no nontrivial 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 970101075 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 990100581 from the Russian Foundation of Fundamental Research. x The work of this author was supported by grant OPG7315 from the Natural Sciences and Eng...
Light Edges in DegreeConstrained Graphs
 Discrete Math
, 2004
"... 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 #. ..."
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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 #.
K5free Bound for the class of Planar Graphs
 European J. Comb
"... We define kdiverse 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 kdiverse colouring using at most f(k) colours. Then using this colo ..."
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We define kdiverse 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 kdiverse colouring using at most f(k) colours. Then using this colouring we obtain a K5free 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
Lightness, Heaviness and Gravity
, 2003
"... 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 prope ..."
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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 almostlight graphs and the absolutely heavy graphs. The paper concludes with few open problems.
The 7cycle C_7 is light in the family of planar Graphs With Minimum degree 5
, 2002
"... A connected graph H is said to be light in the class of graphs H if there exists a positive integer k such that each graph G 2 H that contains an isomorphic copy of H contains a subgraph K isomorphic to H that satis es the inequality P v2V (K) deg G (v) k. It is known that an rcycle Cr is l ..."
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A connected graph H is said to be light in the class of graphs H if there exists a positive integer k such that each graph G 2 H that contains an isomorphic copy of H contains a subgraph K isomorphic to H that satis es the inequality P v2V (K) deg G (v) k. It is known that an rcycle Cr is light in the family of planar graphs with minimum degree 5 if 3 r 6, and not light for r 11. We prove that C7 is also light in this family.