仙人掌圖的（3,1）－全標號

Abstract

Let G be a graph. A (p,1)-total labeling of G is an assignment of integers to each vertex and edge of G such that any adjacent vertices of G are labeled with distinct integers, any 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 of G is the maximum difference between two labels. The minimum span of (p,1)-total labeling of G is called the (p,1)-total number and denote by λ_p^T(G). A cactus graph is a connected graph in which every block is either edge or a cycle. In this thesis, we focus on the (3,1)-total labeling for the class of cactus graphs containing finite cycles joined with a common cut-vertex and show that for any cactus graph G in this class, λ_3^T(G)=Δ+2 where Δ is the maximum degree of G.