 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.39
URN: urn:nbn:de:0030-drops-104439
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10443/
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### Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices

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### 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.

### BibTeX - Entry

```@InProceedings{fulek_et_al:LIPIcs:2019:10443,
author =	{Radoslav Fulek and Jan Kyncl},
title =	{{Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices}},
booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
pages =	{39:1--39:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-104-7},
ISSN =	{1868-8969},
year =	{2019},
volume =	{129},
editor =	{Gill Barequet and Yusu Wang},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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