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Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices

Authors Radoslav Fulek , Jan Kynčl

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Author Details

Radoslav Fulek
  • IST Austria, Am Campus 1, Klosterneuburg 3400, Austria
Jan Kynčl
  • Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00 Praha 1, Czech Republic

Funding

  • Fulek, Radoslav: The author gratefully acknowledges support from Austrian Science Fund (FWF): M2281-N35.
  • Kynčl, Jan: Supported by project 19-04113Y of the Czech Science Foundation (GA\texorpdfstringČ{C}R), by the PRIMUS/17/SCI/3 project of Charles University and by Charles University project UNCE/SCI/004.

Acknowledgements

We are grateful to anonymous referees for comments that helped us to improve the presentation of the results.

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Radoslav Fulek and Jan Kynčl. Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.39

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 Štefankovič 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graphs and surfaces
  • Mathematics of computing → Computations on matrices
Keywords
  • graph genus
  • minimum rank of a partial matrix
  • Hanani-Tutte theorem
  • graph amalgamation

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