On λ-coloring split, chordal bipartite and weakly chordal graphs

A λ-coloring, or L(2, 1)-coloring, of a graph is an assignment of nonnegative integers to its vertices such that adjacent vertices get numbers at least two apart, and vertices at distance two get distinct numbers. Given a graph G, λ(G) is the minimum range of colors for which there exists a λ-coloring of G. A conjecture by Griggs and Yeh (SIAM J. Discrete Math. 5 (1992), 586–595) states that λ(G) is at most Δ2, where Δ is the maximum degree of a vertex in G. We prove that this conjecture holds for weakly chordal graphs. Furthermore, we improve the known upper bounds for λ for chordal bipartite graphs and split graphs.