Abstract
The signed graph \({(G,\sigma)}\) is a graph G = (V, E) with a signature \({\sigma}\) from E to sign group {+, −}. A signed graph is orientation embedded in a surface when it is 2-cellular embedded so that the positive cycles are orientation preserving and the negative cycles are orientation reversing. The demigenus of a signed graph \({(G,\sigma)}\) is the minimal Euler genus over all surfaces S in which \({(G,\sigma)}\) can be orientation embedded. Finding the largest demigenus over all signatures on the complete bipartite graph K m,n is an open problem (Archdeacon, in Problems in topological graph theory, an ongoing online list of open problems, http://www.cems.uvm.edu/~darchdea/problems/signknm.htm). In this paper, by introducing diamond product for signed graphs, the largest demigenus over all signatures on the complete bipartite graph K 3,n is determined.
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This study was supported by National Natural Science Foundation of China (No.11301171) and Tianyuan Fund for Mathematics (No.11226284), and supported by Hunan Province Natural Science Fund Projects (No.13JJ4079).
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Lv, S. The Largest Demigenus Over All Signatures on K 3,n . Graphs and Combinatorics 31, 169–181 (2015). https://doi.org/10.1007/s00373-013-1370-4
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DOI: https://doi.org/10.1007/s00373-013-1370-4