正則圖之(p,1)－全標號

Abstract

A (p,1)–total labeling of a graph G is an assignment of V(G)∪E(G) to integers such that any two 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 minimum span of a (p,1)–total labeling of G is called the (p,1)–total number and denoted by λ_p^T(G). In this thesis, we prove the following two results: (1) For each connected Δ–regular graph G, if p>=max{3,Δ}, then G is bipartite if and only if λ_p^T(G)=Δ+p. (2) For each connected Δ–regular graph G, if p>=max{4,Δ} and G is Class 1, then χ(G)=3 if and only if λ_p^T(G)=Δ+p.