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Nonexistence of almost Moore digraphs of degrees 4 and 5 with self-repeats López Lorenzo, Nacho Messegué Buisan, Arnau Miret Biosca, Josep Maria Universitat Politècnica de Catalunya. Departament de Ciències de la Computació Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs Directed graphs Grafs dirigits An almost Moore (d,k)-digraph is a regular digraph of degree d>1, diameter k>1 and order N(d,k)=d+d2+⋯+dk. So far, their existence has only been shown for k=2, whilst it is known that there are no such digraphs for k=3, 4 and for d=2, 3 when k≥3. Furthermore, under certain assumptions, the nonexistence for the remaining cases has also been shown. In this paper, we prove that (4,k) and (5,k)-almost Moore digraphs with self-repeats do not exist for k≥5. Nacho López: Supported in part by grants PID2020-115442RB-I00 and 2021 SGR-00434. Arnau Messegué: Supported in part by grants Margarita Sala and 2021SGR-00434. Josep M. Miret: Supported in part by grants PID2021-124613OB-I00 and 2021 SGR-00434. Peer Reviewed Postprint (published version) 2023-03-24 Article López, N.; Messegue, A.; Miret, J. Nonexistence of almost Moore digraphs of degrees 4 and 5 with self-repeats. "Electronic journal of combinatorics", 24 Març 2023, vol. 30, núm. 1, article P1.56, p. 1-15. 1077-8926 https://hdl.handle.net/2117/386781 10.37236/11335 eng https://www.combinatorics.org/ojs/index.php/eljc/article/view/v30i1p56 http://creativecommons.org/licenses/by-nd/4.0/ Open Access Attribution-NoDerivatives 4.0 International 15 p. application/pdf