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: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10443/
Fulek, Radoslav ;
Kyncl, Jan
Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices
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},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10443},
URN = {urn:nbn:de:0030-drops-104439},
doi = {10.4230/LIPIcs.SoCG.2019.39},
annote = {Keywords: graph genus, minimum rank of a partial matrix, Hanani-Tutte theorem, graph amalgamation}
}
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Keywords: |
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graph genus, minimum rank of a partial matrix, Hanani-Tutte theorem, graph amalgamation |
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Collection: |
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35th International Symposium on Computational Geometry (SoCG 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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11.06.2019 |
