Let L be an n-element finite lattice. We prove that if L has strictly more than 2 n−5 congruences, then L is planar. This result is sharp, since for each natural number n ≥ 8, there exists a non-planar lattice with exactly 2 n−5 congruences. 7 2 · 2 n−5 , respectively. Since both [2] and Kulin and Mureşan [11] described the lattices witnessing these numbers, it follows from these two papers that |Con(L)| ≥ 7 2 · 2 n−5 implies the planarity of L. So, [2], [11], and even their precursor, Mureşan [12] have naturally lead to the conjecture that if an 2010 Mathematics Subject Classification. 06B10 version July 22, 2018.
For a graph G, let V(G) denote its vertex set and E(G) its edge set. Here we shall only consider reflexive graphs, that is graphs in which every vertex is adjacent to itself. These adjacencies, i.e., the loops, will not be depicted in the figures, although we always assume them present. For graphs G and H, an edge-preserving map (or homomorphism) of G to H is a mapping of V(G) to V(H) such that f(g) is adjacent to f(g′) in H whenever g is adjacent to g′ in G. Because our graphs are reflexive, an edge-preserving map can identify adjacent vertices, i.e., possibly f(g) = f(g′) for some g adjacent to g′, cf. Figure 1(a).
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