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

中文翻译：

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无$$ \ {P_5，K_ {2,3}，K ^ + _ {2,3} \} $$ {P5，K2,3，K2,3 +}无图的加权顶点着色的计算复杂度

在本文中，我们证明了加权顶点着色问题可以在多项式上针对\（\ {P_5，K_ {2,3}，K ^ + _ {2,3} \} \ ） -无图。作为推论，这一事实意味着无权\（\ {P_5，K_ {2,3}，K ^ + _ {2,3} \} \）图的未加权顶点着色问题的多项式时间可解性。像往常一样，\（P_5 \）和\（K_ {2,3} \）分别代表5个顶点上的简单路径以及具有2和3个顶点部分的双斜线，\（K ^ + _ { 2,3} \）表示从\（K_ {2,3} \）通过将其3度顶点与边连接而获得的图。