On decomposing regular graphs into locally irregular subgraphs

Abstract

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. We characterize all connected graphs which cannot be decomposed into locally irregular subgraphs. These are all of odd size and include paths, cycles and a special class of graphs of maximum degree 3. Moreover we conjecture that apart from these exceptions all other connected graphs can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this statement to hold for all regular graphs of degree at least 107. We also support this conjecture by showing that decompositions into three or two such subgraphs might be indicated e.g. for some bipartite graphs (including trees), complete graphs and cartesian products of graphs with this property (hypercubes for instance). We also investigate a total version of this problem, where in some sense also the vertices are being prescribed to particular subgraphs of a decomposition. Both the concepts are closely related to the known 1-2-3 Conjecture and 1-2 Conjecture, respectively, and other similar problems concerning edge colourings. In particular, we improve the result of Addario-Berry et al. (2005) in the case of regular graphs.