Efficient 2-Distance Coloring Method for Maximum and Average Sparse Graphs in Resource Allocation Optimization of Microgrids

In microgrids management, optimizing resource allocation and enhancing efficiency are critical challenges. This paper introduces the 2-distance coloring technique as a novel approach to address these issues by focusing on the coloring problem of sparse graphs. Through rigorous theoretical derivation, we establish the minimum number of colors required for 2-distance coloring in sparse graph scenarios. Specifically, we demonstrate that for a graph G with an average maximum degree less than 2+17/20 and a maximum degree of 6, it can be list 2-distance colored using no more than 11 colors. This means that, given any set of 11 colors for each vertex, a valid 2-distance coloring can be achieved, ensuring no two adjacent vertices share the same color. This finding is pivotal, as it sets a precedent for the number of resources needed to efficiently manage and allocate resources in microgrids systems through 2-distance coloring. The application of this technique in microgrids promises to revolutionize resource distribution, reducing overlap and maximizing efficiency across the network.