On the number of irregular assignments on a graph

Abstract

Let G be a simple graph which has no connected components isomorphic to K 1 or K 2, and let Z + be the set of positive integers. A function ω: E(G)→ Z + is called an assignment on G, and for an edge e of G, ω( e) is called the weight of e. We say that w is of strength s if s = max{ ω( e): e ϵ E( G)}. The weight of a vertex in G is defined to be the sum of the weights of its incident edges. We call assignment w irregular if distinct vertices have distinct weights. Let Irr( G,λ) be the number of irregular assignments on G with strength at most λ. We prove that |Irr(G, λ) − λ q+ c 1λ q−1|= O(λ q−2), λ→∞ where q =| E( G)| and c 1 is a constant depending only on G. An explicit expression for c 1 is given. Analysis of this expression enables us to determine which graph with q edges has the least number of irregular assignments of strength at most λ, for λ sufficiently large.