A smallest graph of girth 10 and valency 3

Distance-Regular Graphs

“…Finally, O'KEEFE & WONG[577] found v(3,1O) = 70; see also BALABAN[23,24]. There are three distinct graphs achieving v (3, 10) = 70 (WONG[794]); these graphs are bipartite.…”

Cages—a survey

“…We arrange these vertices as in Figure 9. If, for any vertex X, the two "opposite" vertices A and B (with respect to X ) are not adjacent, then the number of 7-cycles in this graph is When v -1 is a prime power, the only known minimal ( v , 8)-graphs were FIGURE 12 obtained from projective geometry by Singleton Figure 15 is given in [48]. Recently, Wong [69] has shown that these are the only three possible (3, 10)-cages.…”

On upper bounds of odd girth cages

“…The question of the construction of graphs with small excess is a difficult one, in the papers [1,[3][4][5]7,10,8,11,13,15,[18][19][20][21][22]24,27,26] the authors focused on constructing the smallest one. More details about constructions on cages can be found in the surveys by Wong [28], by Holton and Sheehan [17] or in the recent survey by Exoo and Jajcay [14].…”