# Girth and diameter of a graph with minimum degree at least 3

The problem is motivated by generalizing Moore graphs, graphs with maximum possible girth ($$2\text{diam}+1$$) given the diameter.

Question. Does there exist a graph $$G$$ with $$\text{g}(G)-\text{diam}(G)>8$$ and minimum degree at least $$3$$, where $$\text{g}(G)$$ and $$\text{diam}(G)$$ are the girth and diameter of $$G$$, respectively?

Remark. There's an infinite class of graphs with $$\text{g}(G)-\text{diam}(G)=8$$: the (point,line) incidence graphs of generalized octagons have $$\text{g}(G)=16$$ and $$\text{diam}(G)=8$$.

Bonus points (+200 bounty) for a proof that $$\text{g}-\text{diam}$$ is unbounded for graphs with minimum degree at least $$3$$.

• Have you checked any of the cubic cages in here for whether they give you examples? – M. Winter Mar 20 '20 at 9:38
• Maybe relevant: mathoverflow.net/q/145045 – M. Winter Mar 20 '20 at 9:46
• The graphs of girth 19, 20 and 22 have diameters 14, 15 and 16 respectively. The graph of girth 21 is missing. – LeechLattice Mar 20 '20 at 13:51