Coloring the square of graphs whose maximum average degree is less than 4

The square G 2 of a graph G is the graph defined on V ( G ) such that two vertices u and v are adjacent in G 2 if the distance between u and v in G is at most 2. The maximum average degree of G , m a d ( G ) , is the maximum among the average degrees of the subgraphs of G .It is known in Bonamy et?al. (2014) that there is no constant C such that every graph G with m a d ( G ) < 4 has ? ( G 2 ) ? Δ ( G ) + C . Charpentier (2014) conjectured that there exists an integer D such that every graph G with Δ ( G ) ? D and m a d ( G ) < 4 has ? ( G 2 ) ? 2 Δ ( G ) . Recent result in Bonamy et?al. (2014) 2 implies that ? ( G 2 ) ? 2 Δ ( G ) if m a d ( G ) < 4 - 1 c with Δ ( G ) ? 40 c - 16 .In this paper, we show for an integer c ? 2 , if m a d ( G ) < 4 - 1 c and Δ ( G ) ? 14 c - 7 , then ? ? ( G 2 ) ? 2 Δ ( G ) , which improves the result in Bonamy et?al. (2014) 2. We also show that for every integer D , there is a graph G with Δ ( G ) ? D such that m a d ( G ) < 4 , and ? ( G 2 ) = 2 Δ ( G ) + 2 , which disproves Charpentier's conjecture. In addition, we give counterexamples to Charpentier's another conjecture in Charpentier (2014), stating that for every integer k ? 3 , there is an integer D k such that every graph G with m a d ( G ) < 2 k and Δ ( G ) ? D k has ? ( G 2 ) ? k Δ ( G ) - k .