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 $\text{Z}$⁺ be the set of positive integers. A function $\text{ω: E(G)→}\text{Z}{}^{\mathrm{+}}$ 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

$$\text{|Irr(G, λ) − λ}{}^{\mathrm{q}}\text{+ c}{}_1\text{λ}{}^{\mathrm{q-1}}\text{|= O(λ}{}^{\mathrm{q-2}}\text{), λ→∞}$$

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.