正則圖之（ p，１）－全標號

Abstract

Let G=(V,E) be a graph. A (p,1)-total labeling of G is a mapping form V(G)∪E(G) into {0,…,λ} for some integer λ such that any adjacent vertices of G are labeled with distinct integers, any two adjacent edges of G are labeled with distinct integers and a vertex and its incident edge receive integers that differ by at least p in absolute value. The span of a (p,1)-total labeling is the maximum difference between two labels. The (p,1)-total number of a graph G is the minimum span of a (p,1)-total labeling of G, denoted by λ_p^T (G). In this thesis, we prove that for each connected Δ-regular graph G and each integer k≥4, if p≥max{k+1,Δ} and G is Class 1, then χ(G)=k if and only if λ_p^T (G)=Δ+p+k-2.