廣義彼得森圖的(2,1)-全標號

Abstract

A (p,1)-total labeling of a graph G is to be an assignment of V(G)∪E(G) to integers such that: (i) any two adjacent vertices of G receive distinct integers, (ii) any two adjacent edges of G receive distinct integers, and (iii) 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 to be the (p,1)-total number and denoted by λp,T(G). Let n and k be two positive integers. The graph with vertices {u(1),...u(n)} and {v(1),...,v(n)} and edges u(i)u(i+1),u(i)v(i), and v(i)v(i+k), 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 positive integer n≡0 (mod 3), λ2,T(P(n,k))=5 if k is not divisible by 3.