Locally triangular graphs and normal quotients of the n-cube

Abstract

For an integer \(n\ge 2\), the triangular graph has vertex set the 2-subsets of \(\{1,\ldots ,n\}\) and edge set the pairs of 2-subsets intersecting at one point. Such graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every 2-path lies in a unique quadrangle. We refine this result and provide a characterisation of connected locally triangular graphs as halved graphs of normal quotients of n-cubes. To do so, we study a parameter that generalises the concept of minimum distance for a binary linear code to arbitrary automorphism groups of the n-cube.

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Correspondence to Joanna B. Fawcett.

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The author was supported by the Australian Research Council Discovery Project Grant DP130100106 and thanks the referees for helpful observations.

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Fawcett, J.B. Locally triangular graphs and normal quotients of the n-cube. J Algebr Comb 44, 119–130 (2016). https://doi.org/10.1007/s10801-015-0659-1

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Keywords

  • Locally triangular graph
  • Rectagraph
  • Normal quotient
  • n-cube
  • Semibiplane