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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Michal Staš
Technical University of Košice
Electrical & Electronics Engineering

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On the crossing number of join product of the discrete graph with special graphs of order five Michal Staš
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 8, No 2 (2020): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2020.8.2.10

Abstract

The main aim of the paper is to give the crossing number of join product G+Dn for the disconnected graph G of order five consisting of the complete graph K4 and of one isolated vertex. In the proofs,  it will be extend the idea of the minimum numbers of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph G onto the set of subgraphs which cross the edges of the graph G exactly one times. All methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn for two new graphs.
On the problems of CF-connected graphs Michal Staš; Juraj Valiska
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 11, No 2 (2023): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2023.11.2.12

Abstract

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the plane, and the optimal drawing of G is any drawing at which the desired minimum number of crossings is achieved. We conjecture that a complete graph Kn is CF-connected if and only if it does not contain a subgraph of K8, where a connected graph G is 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 establish the validity of this Conjecture for the complete graphs Kn for any n ≤ 12, and by assuming the Harary-Hill’s Conjecture that cr(Kn)=H(n)=1/4⌊n/2⌋⌊n − 1/2⌋⌊n − 2/2⌋⌊n − 3/2⌋ is also valid for all n > 12. The proofs of this paper are based on the idea of a new concept of a crossing sequence.
Co-Authors Juraj Valiska