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

Abstract

The square G2 of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. The maximum average degree of G, mad(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 mad(G)<4 has χ(G2)≤Δ(G)+C. Charpentier (2014) conjectured that there exists an integer D such that every graph G with Δ(G)≥D and mad(G)<4 has χ(G2)≤2Δ(G). Recent result in Bonamy et al. (2014) [2] implies that χ(G2)≤2Δ(G) if mad(G)<4−1c with Δ(G)≥40c−16.In this paper, we show for an integer c≥2, if mad(G)<4−1c and Δ(G)≥14c−7, then χℓ(G2)≤2Δ(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 mad(G)<4, and χ(G2)=2Δ(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 Dk such that every graph G with mad(G)<2k and Δ(G)≥Dk has χ(G2)≤kΔ(G)−k.