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• W2236586134 abstract "We introduce a catalog of a catalog of a class of (3, g) graphs for even girth g, satisfying 6 <= g <= 16. A (k, g) graph is a graph with regular degree k and girth g. Our catalog has the following properties. Firstly, our catalog contains the smallest known (3, g) graphs. We have identified an appropriate class of trivalent graphs for our catalog, such that the (3, g) graph of minimum order within the class is also the smallest known (3, g) graph. Secondly, our catalog contains (3, g) graphs for more orders than other listings. Thirdly, the class of graphs have been defined so that a practical algorithm to generate graphs can be created, and have implemented such an algorithm. Fourthly, our catalog is infinite, we extend the results into knowledge about infinitely many graphs. Our findings are as follows. Firstly, we have identified Hamiltonian bipartite graphs (HBGs), as a promising class of trivalent graphs that can lead to a catalog of (3, g) graphs for even girth g with graphs for more orders than other listings, that is also expected to contain a (3, g) graph with minimum order. Secondly, our catalog of (3, g) graphs has graphs for more orders compared with other lists, with many graphs outside of the vertex-transitive class. Thirdly, in order to make the computation more tractable, we have introduced the symmetry factor b, which reflects the extent of rotational symmetry. We introduce the D3 chord index notation, a concise notation for trivalent HBGs. Fourthly, results on the minimum order for existence of a (3, g) HBG, and minimum symmetry factor for existence of a (3, g) HBG are of wider interest from an extremal graph theory perspective. Lastly from a cage problem perspective, an analysis on upper bounds and lower bounds within subclasses has been done, where proofs of emptiness of several subclasses partially supports the likeliness of the (3, 14) record graph to be a cage." @default.
• W2236586134 created "2016-06-24" @default.
• W2236586134 creator A5000961052 @default.
• W2236586134 date "2016-01-01" @default.
• W2236586134 modified "2023-09-27" @default.
• W2236586134 title "About the catalog of (3, g) Hamiltonian bipartite graphs" @default.
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