The circular code theory proposes that genes are constituted of two trinucleotide codes: the classical genetic code with 61 trinucleotides for coding the 20 amino acids (except the three stop codons { TAA , TAG , TGA }) and a circular code based on 20 trinucleotides for retrieving, maintaining and synchronizing the reading frame. It relies on two main results: the identification of a maximal C ³ self-complementary trinucleotide circular code X in genes of bacteria, eukaryotes, plasmids and viruses (Michel 2015 J. Theor. Biol. 380, 156–177. ( doi:10.1016/j.jtbi.2015.04.009 ); Arquès & Michel 1996 J. Theor. Biol. 182, 45–58. ( doi:10.1006/jtbi.1996.0142 )) and the finding of X circular code motifs in tRNAs and rRNAs, in particular in the ribosome decoding centre (Michel 2012 Comput. Biol. Chem. 37, 24–37. ( doi:10.1016/j.compbiolchem.2011.10.002 ); El Soufi & Michel 2014 Comput. Biol. Chem. 52, 9–17. ( doi:10.1016/j.compbiolchem.2014.08.001 )). The univerally conserved nucleotides A1492 and A1493 and the conserved nucleotide G530 are included in X circular code motifs. Recently, dinucleotide circular codes were also investigated (Michel & Pirillo 2013 ISRN Biomath. 2013, 538631. ( doi:10.1155/2013/538631 ); Fimmel et al. 2015 J. Theor. Biol. 386, 159–165. ( doi:10.1016/j.jtbi.2015.08.034 )). As the genetic motifs of different lengths are ubiquitous in genes and genomes, we introduce a new approach based on graph theory to study in full generality n -nucleotide circular codes X , i.e. of length 2 (dinucleotide), 3 (trinucleotide), 4 (tetranucleotide), etc. Indeed, we prove that an n -nucleotide code X is circular if and only if the corresponding graph is acyclic. Moreover, the maximal length of a path in corresponds to the window of nucleotides in a sequence for detecting the correct reading frame. Finally, the graph theory of tournaments is applied to the study of dinucleotide circular codes. It has full equivalence between the combinatorics theory (Michel & Pirillo 2013 ISRN Biomath. 2013, 538631. ( doi:10.1155/2013/538631 )) and the group theory (Fimmel et al. 2015 J. Theor. Biol. 386, 159–165. ( doi:10.1016/j.jtbi.2015.08.034 )) of dinucleotide circular codes while its mathematical approach is simpler.

Yury Borisovich Rumer was one of the most important theoretical physicists of the former Soviet Union in the early 1930s. However, he also wrote a few ‘biological papers’ on the standard genetic code after he read Crick's and Nirenberg's pioneering papers on the topic. Rumer's articles on the ‘Systematization of Codons in the Genetic Code’ (Rumer 1966 Doklady Akademii nauk SSSR 167, 1393–1394); Rumer 1968 Doklady Akademii nauk SSSR 183, 225–226; Rumer 1969 Doklady Akademii nauk SSSR 187, 937–938, where he suggested the idea of partitioning codons depending on their redundancy—the first mention of symmetry in the genetic code—were published in Russian only. Due to their importance and their frequent citation, we here present translations of these articles into English in order to make them accessible to a broader community.