Regular Graphs Of Girth Research Articles

Net structure and cages

A g cage of valency k is a regular graph of girth g and valency k with the minimum number of vertices consistent with this condition [12]. An embeddable g net of valency k corresponds to a graph of girth g which can be drawn on a surface in such a way that every face is bounded by a circuit of length g. Similarities between g nets and known g cages are investigated and a possible 5 cage of valency 6 is produced. An existence theorem for regular g nets is also given.

On Hamiltonian Regular Graphs of Girth Six

On Hamiltonian Regular Graphs of Girth Six

On the Non-Existence of a Type of Regular Graphs of Girth 5

ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that1.1Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.