22 research outputs found
Locally Hamiltonian graphs and minimal size of maximal graphs on a surface
We prove that every locally Hamiltonian graph with vertices and
possibly with multiple edges has at least edges with equality if and
only if it triangulates the sphere. As a consequence, every edge-maximal
embedding of a graph graph on some 2-dimensional surface (not
necessarily compact) has at least edges with equality if and only if
also triangulates the sphere. If, in addition, is simple, then for each
vertex , the cyclic ordering of the edges around on is the same
as the clockwise or anti-clockwise orientation around on the sphere. If
contains no complete graph on 4 vertices and has at least 4 vertices, then the
face-boundaries are the same in the two embeddings.Comment: 8 page
Extremal Graphs without Topological Complete Subgraphs
The exact values of the function are known for (see [Cera, Diánez, and Márquez, SIAM J. Discrete Math., 13 (2000), pp. 295--301]), where is the maximum number of edges of a graph of order n not containing a subgraph homeomorphic to the complete graph of order In this paper, for we characterize the family of extremal graphs i.e., the family of graphs with n vertices and edges not containing a subgraph homeomorphic to the complete graph of order $p.
An advance in infinite graph models for the analysis of transportation networks
This paper extends to infinite graphs the most general extremal issues, which are problems of determining the maximum
number of edges of a graph not containing a given subgraph. It also relates the new results with the corresponding situations
for the finite case. In particular, concepts from ‘finite’ graph theory, like the average degree and the extremal number, are
generalized and computed for some specific cases. Finally, some applications of infinite graphs to the transportation of
dangerous goods are presented; they involve the analysis of networks and percolation thresholds.Unión Europea FEDER G-GI3003/IDI