Census of cubic edge-transitive graphs and their optimal LCF codes

This dataset is a complete census of cubic edge-transitive graphs on at most 10,000 vertices - all 4858 of them - in the compact sparse6 graph format. As a bonus, it also includes optimal LCF codes for these graphs (except for two well knwon non-Hamiltonian exceptions, the Petersen graph and the Coxeter graph), giving a human-readable, maximally symmetric description of a Hamiltonian cycle in each. It accompanies the census website fostercensus.graphsym.net. Background. A graph is cubic if every vertex has exactly three neighbours, and edge-transitive if its symmetries can map any edge to any other. A cubic edge-transitive graph is either arc-transitive (symmetries also act transitively on vertices) or semisymmetric (edge-transitive but not vertex-transitive). These form a classical family; the modern census up to 10,000 vertices is due to Conder and Potočnik. The graphs (sparse6). CubicETsparse6.zip contains the whole census as files named CAT_n.s6 and CSS_n.s6 (one per order n), holding the arc-transitive and semisymmetric graphs of that order, one graph per line. sparse6 is a standard compact ASCII encoding read directly by nauty and most graph software (SageMath, NetworkX, …); these codes were generated by nauty 2.8.8. A graph is identified by its order n and an index i, written CAT(n,i) or CSS(n,i). Census summary: 3815 cubic arc-transitive graphs and 1043 cubic semisymmetric graphs, 4858 in total. All are Hamiltonian except two famous cases: the Petersen graph (CAT(10,1)) and the Coxeter graph (CAT(28,1)). Optimal LCF codes. For each Hamiltonian graph we also give an LCF code displaying the largest possible rotational symmetry of a Hamiltonian cycle (the Hamiltonian compression factor). To form it, place the n vertices around a circle in the order the cycle visits them, labelled 0,…,n−1, and join consecutive labels; this uses two of the three edges at each vertex, leaving one "chord". The LCF notation lists, for each vertex i, the offset s to its chord neighbour i+s (mod n, with −n/2 < s ≤ n/2): [s0, …, sn−1]. This single list reconstructs the graph (the neighbours of i are i−1, i+1, i+si). When the offsets repeat with period p, the code is written [s0,…,sp−1]k, where k = n/p is the compression factor. LCF files. CAT_lcf.txt (3815 lines) and CSS_lcf.txt (1043 lines), one graph per line in the format <status> <n> <i>: <LCF code>. For example, ! 24 1: [-5,5,-9,7,-7,9]^4 is CAT(24,1) on 24 vertices, optimal code of period 6, compression factor 4. The status character is ! (provably optimal compression factor), ? (a lower bound believed optimal; smallest such case has 2304 vertices), or # (no Hamiltonian cycle, no code - only Petersen and Coxeter). The indices (n,i) match the sparse6 files. The LCF codes were computed by an algorithm exploiting graph symmetry to find maximally symmetric Hamiltonian cycles, described in the accompanying paper. The implementation of the algorithm is available at the github page. Semiregular automorphisms. The files CubicATsemiregAut.zip and CubicSSsemiregAut.zip contain the information about the semiregular (without fixed points) automorphisms of the graphs in this census, one generator for each representative of the conjugacy classes of semiregular cyclic groups of automorphisms. This precomputed list (obtained with a magma program also available at github page) is needed as the input for the program that computes the optimal LCF codes mentioned in the previous paragraph. After unzipping, the format of each of the two files is as follows: For each graph in the list, the file contains: a line containing 'n,m,k': the number of vertices, the index of the graph among the graphs on n vertices, and the number of automorphisms. the graph, given as an adjacency list of the fom 'n:a,b,c;d,e,f;...', where a,b,c are the neighbours of vertex 1, d,e,f are the neighbours of vertex 2, and so on. k lines, each containing a comma-separated permutation of the vertices in the tabular form representing a semiregular automorphism. Status summary. Of the 3815 arc-transitive graphs, 3715 have provably optimal codes, 98 give lower bounds, and 2 are non-Hamiltonian. All 1043 semisymmetric graphs have provably optimal codes. Please cite: M. Conder, P. Potočnik, Edge-transitive cubic graphs: analysis, cataloguing and enumeration, J. Algebra 685 (2026), 703–737; M. Conder, G. Potočnik, P. Potočnik, Computing the Hamiltonian compression factor of cubic graphs, arXiv:2606.21941 [math.CO].