Papers › Computational methods for finding bi-regular cages
Computational methods for finding bi-regular cages
Jan Goedgebeur, Jorik Jooken, Tibo Van den Eede
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An ({r,m};g)-graph is a (simple, undirected) graph of girth g≥3 with vertices of degrees r and m where 2 ≤r < m . Given r,m,g, we seek the ({r,m};g)-graphs of minimum order, called ({r,m};g)-cages or bi-regular cages, whose order is denoted by n({r,m};g). In this paper, we use computational methods for finding ({r,m};g)-graphs of small order. Firstly, we present an exhaustive generation algorithm, which leads to x2013 previously unknown x2013 exhaustive lists of ({r,m};g)-cages for 24 different triples (r,m,g). This also leads to the improvement of the lower bound of n({4,5};7) from 66 to 69. Secondly, we improve 49 upper bounds of n({r,m};g) based on constructions that start from r-regular graphs. Lastly, we generalize a theorem by Aguilar, Araujo-Pardo and Berman [arXiv:2305.03290, 2023], leading to 73 additional improved upper bounds.
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