Nonexistence of certain edgegirthregular graphs
Abstract
Edgegirthregular graphs (abbreviated as \emph{egr} graphs) are regular graphs in which every edge is contained in the same number of shortest cycles. We prove that there is no $3$regular \emph{egr} graph with girth $7$ such that every edge is on exactly $6$ shortest cycles, and there is no $3$regular \emph{egr} graph with girth $8$ such that every edge is on exactly $14$ shortest cycles. This was conjectured by Goedgebeur and Jooken. A few other unresolved cases are settled as well.
 Publication:

arXiv eprints
 Pub Date:
 March 2024
 DOI:
 10.48550/arXiv.2403.20049
 arXiv:
 arXiv:2403.20049
 Bibcode:
 2024arXiv240320049D
 Keywords:

 Mathematics  Combinatorics;
 05C35 (Primary) 05C38 (Secondary)
 EPrint:
 17 pages, 4 figures