In the b-coloring problem, we aim to assign colors from a set $C$ to the vertices of a graph $G$ in such a way that adjacent vertices do not receive the same color, and for every $c\in C$ we have a $c$-colored vertex $v$ in $G$ such that every color in $C\setminus \left\{c\right\}$ is assigned to at least one of $v$’s neighbors. It has been shown that b-coloring is NP-complete, so we propose in this article an approach for the problem under integer programming techniques. To this end, we give an integer programming formulation and study the associated polytope. We provide several families of valid inequalities, and analyze facetness conditions for them. Finally, we show computational evidence suggesting that the given inequalities may be useful in a branch-and-cut environment.

在b着色问题中，我们旨在从一组中分配颜色$C$ 到图的顶点 $G$ 以这样的方式，相邻的顶点不会收到相同的颜色，并且对于每个 $C\in C$ 我们有一个 $C$色的顶点 $v$ 在 $G$ 这样每一种颜色 $C\setminus \left\{C\right\}$ 被分配给以下至少之一 $v$的邻居。已经表明b着色是NP完全的，因此我们在本文中提出一种在整数编程技术下解决该问题的方法。为此，我们给出了整数编程公式并研究了相关的多表位。我们提供了几个有效的不平等族，并分析了它们的方面条件。最后，我们显示了计算证据，表明给定的不等式在分支剪切环境中可能有用。