Abstract
Wegner conjectured that if G is a planar graph with maximum degree Δ≥8, then χ(G2)≤32Δ+1. This problem has received much attention, but remains open for all Δ≥8. Here we prove an analogous bound on ω(G2): If G is a plane graph with Δ(G)≥36, then ω(G2)≤⌊32Δ(G)⌋+1. In fact, this is a corollary of the following lemma, which is our main result. If G is a plane graph with Δ(G)≥19 and S is a maximal clique in G2 with |S|≥Δ(G)+20, then there exist x,y,z∈V(G) such that S={w:|N[w]∩{x,y,z}|≥2}.
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