22 research outputs found

    Locally Hamiltonian graphs and minimal size of maximal graphs on a surface

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    We prove that every locally Hamiltonian graph with n≥3n\ge 3 vertices and possibly with multiple edges has at least 3n−63n-6 edges with equality if and only if it triangulates the sphere. As a consequence, every edge-maximal embedding of a graph GG graph on some 2-dimensional surface Σ\Sigma (not necessarily compact) has at least 3n−63n-6 edges with equality if and only if GG also triangulates the sphere. If, in addition, GG is simple, then for each vertex vv, the cyclic ordering of the edges around vv on Σ\Sigma is the same as the clockwise or anti-clockwise orientation around vv on the sphere. If GG 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

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    The exact values of the function ex(n;TKp)ex(n;TK_{p}) are known for ⌈2n+53⌉≤p<n{\lceil \frac{2n+5}{3}\rceil}\leq p < n (see [Cera, Diánez, and Márquez, SIAM J. Discrete Math., 13 (2000), pp. 295--301]), where ex(n;TKp)ex(n;TK_p) is the maximum number of edges of a graph of order n not containing a subgraph homeomorphic to the complete graph of order p.p. In this paper, for ⌈2n+63⌉≤p<n−3,{\lceil \frac{2n+6}{3} \rceil}\leq p < n - 3, we characterize the family of extremal graphs EX(n;TKp),EX(n;TK_{p}), i.e., the family of graphs with n vertices and ex(n;TKp)ex(n;TK_{p}) 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

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    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
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