Weighted coloring on planar, bipartite and split graphs: complexity and approximation

In this subsection, we prove that computing the proper orientation number of a graph is still NP -hard for planar bipartite graphs of bounded degree.. The idea of our reduction

The Coloring Game on Planar Graphs with Large Girth, by a result on Sparse Cactuses.. Cl´ ement

Finally, we study approximation in split graphs and provide a 2-approximation algorithm for the case of distinct probabilities and a polynomial time approximation schema under

For example, the complexity of probabilistic coloring remains open, notably for natural graph- families as: general bipartite graphs (under “small” probabilities), paths and cycles

Thus, using the complexity results known on MinDS, we deduce that Upper Edge Cover is NP-hard in planar graphs of maximum degree 3 [24], chordal graphs [8] (even in undirected

– The decision problem RESTRAINED ITALIAN DOMINATION is NP-complete even for bipartite graphs, chordal graphs and planar graphs with maximum degree five, as proved in Theorem 2.1..

There exists δ > 0 such that every planar triangle-free graph of maximum degree at most four and without separating 4-cycles has fractional chromatic number at most 3 − δ.. Dvoˇr´

For example, the complexity of probabilistic coloring remains open, notably for natural graph- families as: general bipartite graphs (under “small” probabilities), paths and cycles