廣義彼得森圖的(2,1)-全標號

Abstract

A (p,1)- total labeling of G is an assignment of integers to V(G)∪E(G) such that any two adjacent vertices of G receive distinct integers, any two adjacent edges of G receive 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^T_P(G). Let n and k be two positive integers. The graph with vertex sets {u(1),...,u(n)} and {v(1),...,v(n)} and edge sets {u(i)u(i+1)|i=1,2,...,n},{u(i)v(i)|i=1,2,...,n} and {v(i)v(i+k)|i=1,2,...,n;k<n}, where addition is modulo n is called generalized Petersen graph and denoted by P(n,k). In this thesis, we mainly focus on the (2,1)-total labeling of the generalized Petersen graph, and we show that for each pair of positive integer n and k, 1<=n, if n≡0(mod 5) and k≠0(mod 5), then λ^T_2(P(n,k))=5. Moreover, we also prove that λ^T_2(P(n,5))=5 if n≡0(mod 25).