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${session.getAttribute("locale")}5On graphs of defect at most 2
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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 [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 2D. 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 the one of (3,D,−ε)-graphs with D ≥ 2 and 0 ≤ ε ≤ 2 [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).]]>Wed 28 Jan 2015 15:20:36 EST]]>