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

AbstractAn L(2,1)-coloring, or λ-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 G, λ is the minimum range of colors for which there exists a λ-coloring of G. 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 λ 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 chordal bipartite graphs, and for split graphs

Tree loop graphs

Many problems involving DNA can be modeled by families of intervals. However, traditional interval graphs do not take into account the repeat structure of a DNA molecule. In the simplest case, one repeat with two copies, the underlying line can be seen as folded into a loop. We propose a new definition that respects repeats and define loop graphs as the intersection graphs of arcs of a loop. The class of loop graphs contains the class of interval graphs and the class of circular-arc graphs. Every loop graph has interval number 2. We characterize the trees that are loop graphs. The characterization yields a polynomial-time algorithm which given a tree decides whether it is a loop graph and, in the affirmative case, produces a loop representation for the tree.Facultad de Ciencias Exacta