# Z_2-Genus of graphs and minimum rank of partial symmetric matrices

Fulek R, Kyncl J. 2019. Z_2-Genus of graphs and minimum rank of partial symmetric matrices. 35th International Symposium on Computational Geometry (SoCG 2019). SoCG: Symposium on Computational Geometry, LIPIcs, vol. 129, 39. 2019_LIPIcs_Fulek.pdf 628.35 KB

Conference Paper | Published | English

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Department
Series Title
LIPIcs
Abstract
The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Stefankovic proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|<=k+1. For complete bipartite graphs K_{m,n}, with n >= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus. We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest.
Publishing Year
Date Published
2019-06-01
Proceedings Title
35th International Symposium on Computational Geometry (SoCG 2019)
Volume
129
Article Number
39
Conference
SoCG: Symposium on Computational Geometry
Conference Location
Portland, OR, United States
Conference Date
2019-06-18 – 2019-06-21
ISSN
IST-REx-ID

### Cite this

Fulek R, Kyncl J. Z_2-Genus of graphs and minimum rank of partial symmetric matrices. In: 35th International Symposium on Computational Geometry (SoCG 2019). Vol 129. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2019. doi:10.4230/LIPICS.SOCG.2019.39
Fulek, R., & Kyncl, J. (2019). Z_2-Genus of graphs and minimum rank of partial symmetric matrices. In 35th International Symposium on Computational Geometry (SoCG 2019) (Vol. 129). Portland, OR, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPICS.SOCG.2019.39
Fulek, Radoslav, and Jan Kyncl. “Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices.” In 35th International Symposium on Computational Geometry (SoCG 2019), Vol. 129. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. https://doi.org/10.4230/LIPICS.SOCG.2019.39.
R. Fulek and J. Kyncl, “Z_2-Genus of graphs and minimum rank of partial symmetric matrices,” in 35th International Symposium on Computational Geometry (SoCG 2019), Portland, OR, United States, 2019, vol. 129.
Fulek R, Kyncl J. 2019. Z_2-Genus of graphs and minimum rank of partial symmetric matrices. 35th International Symposium on Computational Geometry (SoCG 2019). SoCG: Symposium on Computational Geometry, LIPIcs, vol. 129, 39.
Fulek, Radoslav, and Jan Kyncl. “Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices.” 35th International Symposium on Computational Geometry (SoCG 2019), vol. 129, 39, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, doi:10.4230/LIPICS.SOCG.2019.39.
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### Sources arXiv 1903.08637