Graph neural networks have received increased attention over the past years due to their promising ability to handle graph-structured data, which can be found in many real-world problems such as recommender systems and drug synthesis. Most existing research focuses on using graph neural networks to solve homophilous problems, and not much attention has been paid to heterophily-type problems. In this paper, we propose a graph network model for graph coloring, which is a class of representative heterophilous problems. Different from the message passing in conventional graph networks, we introduce negative message passing into a physics-inspired graph neural network for more effective information exchange in handling graph coloring problems. Moreover, a new term in the objective function taking into account the information entropy of nodes is suggested to increase the uniformity of the color assignment of each node, giving the neural network more chance to choose suitable colors for each node. Therefore, it could avoid the final solution getting stuck into the local optimum. Experimental studies are carried out to compare the proposed graph model with five state-of-the-art algorithms on ten publicly available graph coloring problems and d-regular graphs with up to \(10^4\) nodes, demonstrating the effectiveness of the proposed graph neural network.

在过去几年中，图神经网络由于其处理图结构数据的能力而受到越来越多的关注，这种数据可以在许多现实世界的问题中找到，例如推荐系统和药物合成。现有的研究大多集中在使用图神经网络来解决同质问题，而对异质类型的问题关注不多。在本文中，我们提出了一种用于图着色的图网络模型，这是一类代表性的异质问题。与传统图网络中的消息传递不同，我们将负消息传递引入受物理启发的图神经网络中，以便在处理图着色问题时更有效地进行信息交换。此外，建议在目标函数中考虑节点信息熵的新项，以增加每个节点颜色分配的均匀性，使神经网络有更多机会为每个节点选择合适的颜色。因此，它可以避免最终的解决方案陷入局部最优。实验研究将所提出的图模型与五种最先进的算法在十个公开可用的图着色问题和最多\(10^4\)个节点的d正则图上进行比较，证明了该模型的有效性提出的图神经网络。