On graphs of defect at most 2

Abstract

In this paper we consider the degree/diameter problem, namely, given natural numbers Δ ≥ 2 and D ≥ 1 , find the maximum number N ( Δ , D ) of vertices in a graph of maximum degree Δ and diameter D . In this context, the Moore bound M ( Δ , D ) represents an upper bound for N ( Δ , D ) . Graphs of maximum degree Δ , diameter D and order M ( Δ , D ) , called Moore graphs, have turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree Δ ≥ 2 , diameter D ≥ 1 and order M ( Δ , D ) − ϵ with small ϵ > 0 , that is, ( Δ , D , − ϵ ) -graphs. The parameter ϵ is called the defect. Graphs of defect 1 exist only for Δ = 2 . When ϵ > 1 , ( Δ , D , − ϵ ) -graphs represent a wide unexplored area. This paper focuses on graphs of defect 2 . Building on the approaches developed in Feria-Purón and Pineda-Villavicencio (2010) [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a ( Δ , D , − 2 ) -graph with Δ ≥ 4 and D ≥ 4 is 2 D . Second, and most important, we prove the non-existence of ( Δ , D , − 2 ) -graphs with even Δ ≥ 4 and D ≥ 4 ; this outcome, together with a proof on the non-existence of ( 4 , 3 , − 2 ) -graphs (also provided in the paper), allows us to complete the catalogue of ( 4 , D , − ϵ ) -graphs with D ≥ 2 and 0 ≤ ϵ ≤ 2 . Such a catalogue is only the second census of ( Δ , D , − 2 ) -graphs known at present, the first being that of ( 3 , D , − ϵ ) -graphs with D ≥ 2 and 0 ≤ ϵ ≤ 2 Jørgensen (1992) [14]. Other results of this paper include necessary conditions for the existence of ( Δ , D , − 2 ) -graphs with odd Δ ≥ 5 and D ≥ 4 , and the non-existence of ( Δ , D , − 2 ) -graphs with odd Δ ≥ 5 and D ≥ 5 such that Δ ≡ 0 , 2 ( mod D ) . Finally, we conjecture that there are no ( Δ , D , − 2 ) -graphs with Δ ≥ 4 and D ≥ 4 , and comment on some implications of our results for the upper bounds of N ( Δ , D ) .