Publication Date
3-17-2026
Document Type
Article
Organizational Units
College of Natural Science and Mathematics, Mathematics
Keywords
Path cover, Odd-cover, Linear arboricity, Gallai's conjecture
Abstract
We introduce and study “path odd-covers,” a weakening of Gallai's path decomposition problem and a strengthening of the linear arboricity problem. The path odd-cover number of a graph G is the minimum cardinality of a collection of paths whose vertex sets are contained in and whose symmetric difference of edge sets is. We prove an upper bound on in terms of the maximum degree Δ and the number of odd-degree vertices of the form . This bound is only a factor of 2 from a rather immediate lower bound of the form . We also investigate some natural relaxations of the problem which highlight the connection between the path odd-cover number and other well-known graph parameters. For example, when allowing for subdivisions of G, the previously mentioned lower bound is always tight except in some trivial cases. Further, a relaxation that allows for the addition of isolated vertices to G leads to a match with the linear arboricity when G is Eulerian. Finally, we transfer our observations to establish analogous results for cycle odd-covers.
Copyright Date
3-17-2026
Copyright Statement / License for Reuse
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Rights Holder
Steffen Borgwardt, Calum Buchanan, Eric Culver, Bryce Frederickson, Puck Rombach, and Youngho Yoo
Provenance
Received from Elsevier
File Format
application/pdf
Language
English (eng)
Extent
24
File Size
1.41 MB
Publication Statement
Copyright is held by the Authors. User is responsible for all copyright compliance. This article was originally published as
Borgwardt, S., Buchanan, C., Culver, E., Frederickson, B., Rombach, P., & Yoo, Y. (2026). Path Odd-Covers of Graphs. Discrete Mathematics, 349(9). https://doi.org/10.1016/j.disc.2026.115078
Publication Title
Discrete Mathematics
Volume
349
Issue
9
First Page
115078
ISSN
0012-365X
Recommended Citation
Borgwardt, Steffen; Buchanan, Calum; Culver, Eric; Frederickson, Bryce; Rombach, Puck; and Yoo, Youngho, "Path Odd-Covers of Graphs" (2026). Mathematics: Faculty Scholarship. 60.
https://digitalcommons.du.edu/math_faculty/60
https://doi.org/10.1016/j.disc.2026.115078