The Computational Complexity Of Weighted Vertex Coloring For {P-5,K-2,K-3,K-2,3(+)}-Free Graphs

In this paper, we show that the weighted vertex coloring problem can be solved in polynomial on the sum of vertex weights time for {P5,K2,3,K2,3+}{P_5,K_{2,3}, K<^>+_{2,3}\}$$\end{document}-free graphs. As a corollary, this fact implies polynomial-time solvability of the unweighted vertex coloring problem for {P5,K2,3,K2,3+}P_5,K_{2,3},K<^>+_{2,3}\}$$\end{document}-free graphs. As usual, P5 and K2,3 stands, respectively, for the simple path on 5 vertices and for the biclique with the parts of 2 and 3 vertices, K2,3+ denotes the graph, obtained from a K2,3 by joining its degree 3 vertices with an edge.