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We prove that each polyhedral triangular face free map G on a compact 2-dimensional manifold M with Euler characteristic χ ( 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+...
Suppose χ ( G )= r and P ⊆ V ( G ). It is known that if the distance between any two vertices in P is at least 4, then any ( r +1)-coloring of P extends to an ( r +1)-coloring of all of G , but an r -coloring of P might not extend to an r -coloring of G . We show that if the distance between...
In this paper we generalize the concept of alternating knots to alternating graphs and show that every abstract graph has a spatial embedding that is alternating. We also prove that every spatial graph is a subgraph of an alternating graph. We define n -composition for spatial graphs and...
Let G be a connected k -regular vertex-transitive graph on n vertices. For S ⊆ V ( G ) let d ( S ) denote the number of edges between S and V ( G )\ S . We extend results of Mader and Tindell by showing that if d ( S )< 2 9 ( k +1) 2 for some S ⊆ V ( G ) with 1 3 ( k +1)⩽| S |⩽ 1 2 n ,...
In this paper we study the question of existence of a basis consisting only of cycles for the lattice Z ( M ) generated by the cycles of a binary matroid M . We show that if M has no Fano dual minor, then any set of fundamental circuits can be completed to a cycle basis of Z ( M ); moreover, for...
A graph is called weakly pancyclic if it contains cycles of all lengths between its girth and circumference. A substantial result of Häggkvist, Faudree, and Schelp (1981) states that a Hamiltonian non-bipartite graph of order n and size at least ⌊( n −1) 2 /4⌋+2 contains cycles of every...
We show how to construct all the graphs that can be embedded on both the torus and the Klein bottle as their triangulations.
Dans cet article, nous explicitons la forme générale des séries génératrices des cartes pointées sur une surface orientable, énumérées en fonction des nombres de sommets et de faces. Toutes ces séries génératrices sont des séries rationnelles d'un unique système paramétrique des...
The main result of this series serves to reduce several problems about general graphs to problems about graphs which can “almost” be drawn in surfaces of bounded genus. In applications of the theorem we usually need to encode such a nearly embedded graph as a hypergraph which can be drawn...
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