On bipartite graphs of defect at most 4

• Authors: Ramiro Feria-Purón;Guillermo Pineda-Villavicencio
• Venue: Discrete Applied Mathematics
• Year: 2012

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers @D=2 and D=2, find the maximum number N^b(@D,D) of vertices in a bipartite graph of maximum degree @D and diameter D. In this context, the Moore bipartite bound M^b(@D,D) represents an upper bound for N^b(@D,D). Bipartite graphs of maximum degree @D, diameter D and order M^b(@D,D)-called Moore bipartite graphs-have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree @D=2, diameter D=2 and order M^b(@D,D)-@e with small @e0, that is, bipartite (@D,D,-@e)-graphs. The parameter @e is called the defect. This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs. Bipartite graphs of defect 2 have been studied in the past; if @D=3 and D=3, they may only exist for D=3. However, when @e2 bipartite (@D,D,-@e)-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite (@D,D,-4)-graphs; the complete catalogue of bipartite (3,D,-@e)-graphs with D=2 and 0@?@e@?4; the complete catalogue of bipartite (@D,D,-@e)-graphs with @D=2, 5@?D@?187 (D6) and 0@?@e@?4; a proof of the non-existence of all bipartite (@D,D,-4)-graphs with @D=3 and odd D=5. Finally, we conjecture that there are no bipartite graphs of defect 4 for @D=3 and D=5, and comment on some implications of our results for the upper bounds of N^b(@D,D).