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On the Problems of CF -Connected Graphs for K l,m,n

Author

Listed:
  • Michal Staš

    (Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, 042 00 Košice, Slovakia)

  • Mária Timková

    (Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, 042 00 Košice, Slovakia)

Abstract

A connected graph, G , is Crossing Free -connected ( CF -connected) if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G . We conjecture that a complete tripartite graph, K l , m , n , is CF -connected if and only if it does not contain any of the following as a subgraph: K 1 , 2 , 7 , K 1 , 3 , 5 , K 1 , 4 , 4 , K 2 , 2 , 5 , K 3 , 3 , 3 . We examine the idea that K 1 , 2 , 7 , K 1 , 3 , 5 , K 1 , 4 , 4 , and K 2 , 2 , 5 are the first non- CF -connected complete tripartite graphs. The CF -connectedness of K l , m , n with l , m , n ≥ 3 is dependent on the knowledge of crossing numbers of K 3 , 3 , n . In this paper, we prove various results that support this conjecture.

Suggested Citation

  • Michal Staš & Mária Timková, 2024. "On the Problems of CF -Connected Graphs for K l,m,n," Mathematics, MDPI, vol. 12(13), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:13:p:2068-:d:1427071
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