Abstract:

For an undirected connected graph G(V, E), if there exists a injective function f: V(G)∪E(G)→{1, 2, …, |V|+|E|}, such that any edge $\operatorname{Sum}(u)=f(u)+\sum\limits_{u v \in E(G)} f(u v)=K $, K is a constant, where u is any vertex in G(p, q) with the same degree, then G(p, q) is a vertex reducible total labeling (VRTL) graph, and the injective function f is vertex reducible total labeling. By means of computer and the way of optimizing the solution space of the vertex reducible total labeling, the algorithm of the vertex reducible total labeling is designed, and the solution space of the vertex reducible total labeling is recursively searched. By observing the labeling law of graphs within finite vertices, the labeling law of graphs of the same kind that can describe infinite vertices is extended and the total labeling theorems with extensibility and mathematical proof are given.

Key words: graph, total labeling, reducible total labeling, vertex reducible total labeling, vertex reducible total labeling algorithm