Vertex coloring of a graph is the assignment of labels to the vertices of the graph so that adjacent vertices have different labels. In the case of polyhedral graphs, the chromatic number is 2, 3, or 4. Edge coloring problem and face coloring problem can be converted to vertex coloring problem for appropriate polyhedral graphs. We have been developed an interactive learning system of polyhedra,...

If a graph G contains no subgraph isomorphic to some graph H, then G is called H-free. A coloring of a graph G = (V,E) is a mapping c : V → {1, 2, . . .} such that no two adjacent vertices have the same color, i.e., c(u) 6= c(v) if uv ∈ E; if |c(V )| ≤ k then c is a k-coloring. The Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k. The Precolori...

The (∆+1)-coloring problem is a fundamental symmetry breaking problem in distributed computing. We give a new randomized coloring algorithm for (∆ + 1)-coloring running in O( √ log ∆) + 2O( √ log logn) rounds with probability 1 − 1/nΩ(1) in a graph with n nodes and maximum degree ∆. This implies that the (∆ + 1)-coloring problem is easier than the maximal independent set problem and the maximal...