Dynamic irregular triangulated meshes are used in adaptive grid partial di erential equation (PDE) solvers, and in simulations of random surface models of quantum gravity in physics and cell membranes in biology. Parallel algorithms for random surface simulations and adaptive grid PDE solvers require coloring of the triangulated mesh, so that neighboring vertices are not updated simultaneously....

In this paper we are interested in the elaboration of an approached solution to the sum coloring problem (MSCP), which is an NP-hard problem derived from the graphs coloring (GCP). The problem (MSCP) consists in minimizing the sum of colors in a graph. Our resolution approach is based on an hybridization of a genetic algorithm and a local heuristic based on an improvement of the maximal indepen...

In a proper vertex coloring of a graph a subgraph is colorful if its vertices are colored with different colors. It is well-known that in every proper coloring of a k-chromatic graph there is a colorful path Pk on k vertices. If the graph is k-chromatic and triangle-free then in any proper coloring there is also a path Pk which is an induced subgraph. N.R. Aravind conjectured that these results...

Gimbel and Thomassen asked whether 3-colorability of a triangle-free graph drawn on a fixed surface can be tested in polynomial time. We settle the question by giving a linear-time algorithm for every surface which combined with previous results gives a lineartime algorithm to compute the chromatic number of such graphs. Our algorithm is based on a structure theorem that for a triangle-free gra...

In a recent result, Khot and Saket [FOCS 2014] proved the quasi-NP-hardness of coloring a 2-colorable 12-uniform hypergraph with 2(logn) Ω(1) colors. This result was proved using a novel outer PCP verifier which had a strong soundness guarantee. We reduce the arity in their result by modifying their 12-query inner verifier to an 8-query inner verifier based on the hypergraph coloring hardness r...

In a recent result, Khot and Saket [FOCS 2014] proved the quasi-NP-hardness of coloring a 2-colorable 12-uniform hypergraphwith 2 Ω(1) colors. This result was proved using a novel outer PCP verifier which had a strong soundness guarantee. In this note, we show that we can reduce the arity of their result by modifying their 12-query inner verifier to an 8-query inner verifier based on the hyperg...

Despite the fact that some vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not “under control” from a polyhedral point of view. The equivalence between optimization and polyhedral separation suggests that, for these problems, there must exist formulations admitting some elegant characterization for the polytopes associated to them. Therefo...

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge w...

Global register allocation is one of the most important optimizations in a compiler. Since the early 80’s, register allocation by graph coloring has been the dominant approach. The traditional formulation of graph-coloring register allocation implicitly assumes a single bank of non-overlapping general-purpose registers and does not handle irregular architectural features like overlapping regist...

The work presents some new algorithms realized recently in the package TESTAS. The package decides whether or not DFA is synchronizing, several procedures find relatively short synchronizing words and a synchronizing word of the minimal length. We check whether or not a directed graph has a road coloring that turns the graph into a synchronizing deterministic finite automaton (DFA). The algorit...