Abstract
An edge-girth-regular \(egr(v,k,g,\lambda )\)-graph \(\Gamma \) is a k-regular graph of order v and girth g in which every edge is contained in \(\lambda \) distinct g-cycles. Edge-girth-regularity is shared by several interesting classes of graphs which include edge- and arc-transitive graphs, Moore graphs, as well as many of the extremal k-regular graphs of prescribed girth or diameter. Infinitely many \(egr(v,k,g,\lambda )\)-graphs are known to exist for sufficiently large parameters \((k,g,\lambda )\), and in line with the well-known Cage Problem we attempt to determine the smallest graphs among all edge-girth-regular graphs for given parameters \((k,g,\lambda )\). To facilitate the search for \(egr(v,k,g,\lambda )\)-graphs of the smallest possible orders, we derive lower bounds in terms of the parameters k, g and \(\lambda \). We also determine the orders of the smallest \(egr(v,k,g,\lambda )\)-graphs for some specific parameters \((k,g,\lambda )\), and address the problem of the smallest possible orders of bipartite edge-girth-regular graphs.
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Acknowledgements
The second author partially supported by VEGA 1/0423/20. The third author partially supported by VEGA 1/0423/20, APVV-15-0220, and by the Slovenian Research Agency (research project N1-0038, N1-0062, J1-9108). The fourth author partially supported by Grant of SGS No. SP2020/114, VŠB - Technical University of Ostrava, Czech Republic. The authors also thank Grahame Erskine for providing them with the results of his computer search for 4-regular graphs of girth 4.
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Drglin, A.Z., Filipovski, S., Jajcay, R. et al. Extremal Edge-Girth-Regular Graphs. Graphs and Combinatorics 37, 2139–2154 (2021). https://doi.org/10.1007/s00373-021-02368-9
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DOI: https://doi.org/10.1007/s00373-021-02368-9