Adjacent Vertex Reducible Total Coloring Algorithm for Several Joint Graphs

For a simple graph G(V, E), if there exists a positive integer k (1 ≤ k ≤ |E| + |G|) and a mapping f: V(G) U E(G) → {1, 2, …, k} such that the color sets of adjacent vertices with the same degree are identical, then f represents the Adjacent Vertex Reducible Total Coloring(AVRTC) of the graph G. Building upon the existing research on reducible edge coloring and Adjacent Vertices Reducible Edge Labeling, a new algorithm for Adjacent Vertex Reducible Total Coloring is designed based on practical problems. This algorithm conducts research on several joint graphs through iterative optimization. Analyzing the experimental results obtains the adjacent vertex reducible total chromatic number of some joint graphs, summarizes several theorems, and provides proof.

I. Introduction

The graph coloring problem has always been a classic topic in graph theory [1], with significant value in theoretical research and practical applications. In recent years, with the vigorous development of information technology, it has strongly promoted the progress of graph theory. Many scholars have conducted a series of research and exploration around it, leading to the introduction of numerous new concepts, such as vertex-distinguishing total coloring [2]–[3], edge-magic total labeling, and vertex-magic edge coloring, among others. In addition, with the development of machine learning technology, scholars have also attempted to use techniques such as neural networks [4]–[5] to solve the graph coloring problem. Although still in the exploratory stage, it holds great research prospects.