A generalization of Faudree–Lehel conjecture holds almost surely for random graphs

Abstract

The irregularity strength of a simple graph G=(V,E) , denoted s(G) is a certain measure of the level of irregularity of a graph. It indicates how hard it is to make an irregular multigraph of G via multiplication of its selected edges. It is however more commonly set forth through k‐weightings, that is, mappings ω:E→{1,2,…,k} , assigning every vertex v∈V the weighted degree σ(v):=∑e∋vω(e) . In this setting, s(G) is precisely defined as the least k admitting a k‐weighting of G which attributes pairwise distinct weighted degrees to all vertices of G. It is known that s(G)>n/d in the case of d‐regular graphs with order n and d>1 . An open conjecture of Faudree and Lehel from the 1980s states that s(G)≤n/d+c in turn for some finite constant c independent of d. It is believed that the natural strengthening of this conjecture toward all graphs where d is substituted by the minimum degree δ should also hold. We confirm this supposition in the case of random graphs. Namely, we show that asymptotically almost surely the generalization of Faudree‐Lehel Conjecture holds for a random graph G∈𝒢(n,p) for any constant p, that is, that s(G) takes one of the three values: ⌈n/δ⌉ , ⌈n/δ⌉+1 , or ⌈n/δ⌉+2 . This is implied by the fact that a.a.s. p−1