Edge-partitioning graphs into regular and locally irregular components

A graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graphG, there exists a minimum integer χirr(G) such thatG admits an edge-partition into χ ′ irr(G) classes, each of which induces a locally irregular graph. In particular, they conjectured that χirr(G) ≤ 3 for everyG, unlessG belongs to a well-characterized family of non-decomposable graphs. This conjecture is far from being settled, as notably (1) no constant upper bound on χirr(G) is known for G bipartite, and (2) no satisfactory general upper bound on χ ′ irr(G) is known.