Odd Covers of Graphs

Authors

• Calum Buchanan, University of Vermont
• Alexander Clifton, Institute for Basic Science, Daejeon, South Korea
• Eric Culver, Brigham Young University
• Jiaxi Nie, Fudan University
• Jason O'Neill, California State University, Los Angeles
• Puck Rombach, University of Vermont
• Mei Yin, University of DenverFollow

Publication Date

5-5-2023

Document Type

Article

Organizational Units

College of Natural Science and Mathematics, Mathematics

Keywords

Odd covers, Graphs, Bicliques, Minimum cardinality, Bipartite graphs

Abstract

Given a finite simple graph 𝐺, an odd cover of 𝐺 is a collection of complete bipartite graphs, or bicliques, in which each edge of 𝐺 appears in an odd number of bicliques, and each nonedge of 𝐺 appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of 𝐺 by 𝑏2⁡(𝐺) and prove that 𝑏2⁡(𝐺) is bounded below by half of the rank over 𝔽2 of the adjacency matrix of 𝐺. We show that this lower bound is tight in the case when 𝐺 is a bipartite graph and almost tight when 𝐺 is an odd cycle. However, we also present an infinite family of graphs which shows that this lower bound can be arbitrarily far away from 𝑏2⁡(𝐺). Babai and Frankl proposed the “odd cover problem,” which in our language is equivalent to determining 𝑏2⁡(𝐾𝑛). In this paper, we determine that 𝑏2⁡(𝐾𝑛) is 𝑛 /2 when 8 ∣𝑛 and is (𝑛 +1) /2 when 𝑛 is equivalent to 1 or −1 modulo 8.

Copyright Date

5-5-2023

Copyright Statement / License for Reuse

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Rights Holder

The Authors

Provenance

Received from author

File Format

application/pdf

Language

English (eng)

Extent

20 pgs

File Size

1208 KB

Publication Statement

Copyright is held by the authors. This is an open access article under the terms of the Creative Commons Attribution‐NonCommercial‐NoDerivatives License. Article originally published as: 

Buchanan, C., Clifton, A., Culver, E., Nie, J., O'Neill, J., Rombach, P., & Yin, M. (2023). Odd covers of graphs, Journal of Graph Theory, 104(2), 420-439. https://doi.org/10.1002/jgt.22970

Publication Title

Journal of Graph Theory

Volume

104

Issue

2

First Page

420

Last Page

439

ISSN

1097-0118

Recommended Citation

Buchanan, C., Clifton, A., Culver, E., Nie, J., O'Neill, J., Rombach, P., & Yin, M. (2023). Odd covers of graphs, Journal of Graph Theory, 104(2), 420-439. https://doi.org/10.1002/jgt.22970

DOI Link

https://doi.org/10.1002/jgt.22970