A New Efficient Approximation Algorithm for Chromatic Number

We design a new approximation polynomial-time algorithm for the graph coloring problem. Our proposal is based on selecting, in iterative manner, a critical vertex v of the graph. The criterion to select is based on choose the node with maximum degree and with maximum degree of its neighborhood into the set of vertices composing odd cycles. The algorithm consists of two embedded loops. While in the internal loop a critical node is selected to be colored and it is deleted as well as its upon edges from the current graph. The external loop controls when there is not possible to select more vertices and, while remains odd cycles in the current graph, new colors are used. The stop criterion to finish the two loops is when the current subgraph is bipartite. Our algorithm establishes an average number of ( √ 4δ + 1 + 2δ − 1)/2 colors for approximate the chromatic number of any graph G, δ being the initial average degree of the input graph G.