Cliques in squares of graphs with maximum average degree less than 4

Hocquard, Kim, and Pierron constructed, for every even integer , a 2‐degenerate graph with maximum degree such that . We prove for (a) all 2‐degenerate graphs and (b) all graphs with , upper bounds on the clique number of that match the lower bound given by this construction, up to small additive constants. We show that if is 2‐degenerate with maximum degree , then (with when is sufficiently large). And if has and maximum degree , then . Thus, the construction of Hocquard et al. is essentially the best possible. Our proofs introduce a “token passing” technique to derive crucial information about nonadjacencies in of vertices that are adjacent in . This is a powerful technique for working with such graphs that has not previously appeared in the literature.