On the number of irregular assignments on a graph

Abstract Let G be a simple graph which has no connected components isomorphic to K1 or K2, 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 1 λ q−1 |= O(λ q−2 ), λ→∞ where q =|E(G)| and c1 is a constant depending only on G. An explicit expression for c1 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.