The square of a graph G, denoted G^2, has the same vertex set as G and has an edge between two vertices if the distance between them in G is at most 2. In general, ∆(G) + 1 ≤ χ(G^2) ≤ ∆(G)^2 + 1 for every graph G. Charpentier (2014) asked whether χ(G^2) ≤ 2∆(G) if mad(G) < 4. But Hocquard, Kim, and Pierron (2019) answered his
question negatively. For every even value of ∆(G), they constructed a 2-degenerate graph G such that ω(G^2) = 2.5∆(G). Note that if G is a 2-degenerate graph, then mad(G) < 4. Thus, we have that 2.5 ∆(G) ≤ max{χ(G^2) : G is a 2-degenerate graph} ≤ 3∆(G) + 1.
So, it was naturally asked whether there exists a constant D_0 such that χ(G^2) ≤ 2.5 ∆(G) if G is a 2-degenerate graph with ∆(G) ≥ D_0. Recently Cranston and Yu (2024) showed that ω(G2) ≤ 2.5 ∆(G) + 72 if G is a 2-degenerate graph, and ω(G2) ≤ 2.5 ∆(G) + 60 if G is a 2-degenerate graph with ∆(G) ≥ 1729. We show that there exists a
constant D0 such that ω(G2) ≤ 2.5 ∆(G) if G is a 2-degenerate graph with ∆(G) ≥ D_0. This upper bound on ω(G^2) is tight. This is joint work with Xiaopan Lian (Nankai University)