On L(2,1)L(2,1)-coloring split, chordal bipartite, and weakly chordal graphs

An L(2,1)L(2,1)-coloring, or λλ-coloring, of a graph is an assignment of non-negative 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 GG, λλ is the minimum range of colors for which there exists a λλ-coloring of GG. A conjecture by Griggs and Yeh [J.R. Griggs, R.K. Yeh, Labelling graphs with a condition at distance 2, SIAM Journal on Discrete Mathematics 5 (1992) 586–595] states that λλ is at most Δ2Δ2, where ΔΔ is the maximum degree of a vertex in GG. We prove that this conjecture holds for weakly chordal graphs. Furthermore, we improve the known upper bounds for chordal bipartite graphs, and for split graphs.