Small girth 6 graphs for keyboard wiring

This post is an expanded and reworked version of the table in my first keyboards and Graph Theory blog post, listing the best (in terms of maximal number of edges) girth 6 bipartite small graph(s). I added many of these to the House of Graphs database, and then in October the mathematician Steven Van Overberghe added a few more. My interest here is applying the graphs to the design of diode-free computer keyboards where the bipartite matrix becomes a sparse scanning matrix, and girth 6 becomes 4-key rollover (4KRO).

Nodes, GPIO	Edges, Keys	Girth 6 graph(s)	Example Keyboard(s)
10	12	Unnamed Graph (circle with two crossing lines)	24-key 4x6 macropad
11	14	Unnamed Graph	28-key split Zplitzalp
12	15	Unnamed Graph (circle with three crossing lines)	30-key split Rommana
13	18	Partial Heawood Graph	36-key split 3W6, 34-key split Forager Acid
14	21	Heawood Graph (wikipedia)	42-key split Heawood42, 38-key split Totem
15	22	Unnamed Graph	44-key split
16	24	Möbius–Kantor Graph (wikipedia)	24-key MoKa-NP
17	26	Unnamed Graph	26-key ʻākohekohe, 26-key smallcat
18	29	Unnamed Graph (better than the Pappus Graph etc)	28-key Zilpzalp, 28-key Grumpy, 29-key Fitis
19	31	Unnamed Graph	30 key Hummingbird
20	34	Unnamed Graph	34-key Le Chiffre, 33 or 34-key reviung34, 32-key Visorbearer
21	36	Hesse Configuration Incidence Graph	36-key Gamma Omega Hesse
22	39	Unnamed Graph, Unnamed Graph	39-key Reviung39
23	42	Unnamed Graph	42-key Osprey
24	45	Unnamed Graph (better than Nauru graph etc).	45-key Splaynck, 44-key Masonry
25	48+	Partial PG(2,3)	48-key
26	52	Incidence graph of the projective plane of order 3, PG(2,3) (two keys per GPIO)	52-key Slump52
27	53+	Partial JESK is 52, trivial expansion of PG(2,3) gives 53.	53-key Reviung53
28	56	JESK graph (two keys per GPIO)	56-key JESK56
29	57+	Trivial expansion of JESK gives 57. Reduction of Integral Graph etc gives only 56 (as degree 4).	57-key
30	60	Integral Graph, Unnamed graph, Unnamed graph (two keys per GPIO)	60-key
31	63+	Trivial expansion gives 61+. Reduction can give 63 as degree 4/5.	63-key
32	67	Unnamed cage graph, Unnamed cage graph (over two keys per GPIO)	67-key

Those odd number entries with a plus are where I think a better graph may exist. The even ones tend to be more symmetric, and thus more interesting from a mathematical perspective, and thus more likely to have been added to the House of Graphs database.

The keyboards listed in bold are already using girth 6 graph diode-free designs, three of mine (two built, one in planning), and three from T. G. Marbach. I'm currently working on the "Slump52", a curvy keyboard design intended for a child (or anyone with small hands) combining a split-style layout with dedicated number pad and cursors. More on that later...