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11
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...
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 #.
NEARLY–LIGHT CYCLES IN EMBEDDED GRAPHS AND CROSSING–CRITICAL GRAPHS
"... Abstract. We find a lower bound for the proportion of face boundaries of an embedded graph that are nearly–light (that is, they have bounded length and at most one vertex of large degree). As an application, we show that every sufficiently large k–crossing–critical graph has crossing number at most ..."
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Abstract. We find a lower bound for the proportion of face boundaries of an embedded graph that are nearly–light (that is, they have bounded length and at most one vertex of large degree). As an application, we show that every sufficiently large k–crossing–critical graph has crossing number at most 2k + 23. 1.
Nearlylight cycles in embedded graphs and in crossingcritical graphs, preprint
"... Abstract. We find a lower bound for the proportion of face boundaries of an embedded graph that are nearly–light (that is, they have bounded length and at most one vertex of large degree). As an application, we show that every sufficiently large k–crossing–critical graph has crossing number at most ..."
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Abstract. We find a lower bound for the proportion of face boundaries of an embedded graph that are nearly–light (that is, they have bounded length and at most one vertex of large degree). As an application, we show that every sufficiently large k–crossing–critical graph has crossing number at most 2k + 23. 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.
3POLYTOPES WITH CONSTANT FACE WEIGHT
"... Dedicated toProfessor E.Jucovic on the occasion of his 70th birthday Abstract. The weight of a face in a 3polytope is the sum of degrees of vertices which are incident with. In the present paper we determine the number of di erent regular 3polytopes for which the weight of each face is w 9. In the ..."
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Dedicated toProfessor E.Jucovic on the occasion of his 70th birthday Abstract. The weight of a face in a 3polytope is the sum of degrees of vertices which are incident with. In the present paper we determine the number of di erent regular 3polytopes for which the weight of each face is w 9. In the nonregular case we havesimilar results if 9 w 21 and if 28 w. 1.
Paths Of Low Weight In Planar Graphs
"... The existence of subgraphs of low degree sum of their vertices in planar graphs is investigated. Let K1;3 , a subgraph of a graph G, be an (x; a; b; c)star, a star with a central vertex of degree x and three leaves of degrees a, b and c in G. The main results of the paper are: 1. Every planar g ..."
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The existence of subgraphs of low degree sum of their vertices in planar graphs is investigated. Let K1;3 , a subgraph of a graph G, be an (x; a; b; c)star, a star with a central vertex of degree x and three leaves of degrees a, b and c in G. The main results of the paper are: 1. Every planar graph G of minimum degree at least 3 contains an (x; a; b; c)star with a b c and (i) x = 3, a 10, or (ii) x = 4, a = 4, 4 b 10, or (iii) x = 4, a = 5, 5 b 9, or (iv) x = 4, 6 a 7, 6 b 8, or (v) x = 5, 4 a 5, 5 b 6 and 5 c 7, or (vi) x = 5 and a = b = c = 6.
Heavy Paths, Light Stars, and Big Melons
, 2002
"... A graph H is defined to be light in the family H of graphs if there exists a finite number w(H;H) such that each G 2 H which contains H as a subgraph, contains also a subgraph K = H such that the sum of degrees (in G) of vertices of K (that is, the weight of K in G) is at most w(H;H). In this ..."
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A graph H is defined to be light in the family H of graphs if there exists a finite number w(H;H) such that each G 2 H which contains H as a subgraph, contains also a subgraph K = H such that the sum of degrees (in G) of vertices of K (that is, the weight of K in G) is at most w(H;H). In this paper we study the conditions related to weight of fixed subgraph of plane graphs which can enforce the existence of light graphs in families of plane graphs. For the families of plane graphs and triangulations whose edges are of weight w we study the necessary and sucient conditions for lightness of certain graphs according to values of w.
appropriate 3polytopes. Manuscript
, 2012
"... Abstract: We prove that every normal plane map, as well as every 3polytope, has a path on three vertices whose degrees are bounded from above by one of the following triplets: $(3,3,\infty)$, $(3,4,11)$, $(3,7,6)$, $(3,10,4)$, $(3,15,3)$, $(4,4,9)$, $(6,4,8)$, $(7,4,7)$, and $(6,5,6)$. No parameter ..."
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Abstract: We prove that every normal plane map, as well as every 3polytope, has a path on three vertices whose degrees are bounded from above by one of the following triplets: $(3,3,\infty)$, $(3,4,11)$, $(3,7,6)$, $(3,10,4)$, $(3,15,3)$, $(4,4,9)$, $(6,4,8)$, $(7,4,7)$, and $(6,5,6)$. No parameter of this description can be improved, as shown by