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. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph 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 to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].
Highlights
All graphs considered are simple and finite
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
Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least 1010
Summary
All graphs considered are simple and finite. We follow [6] for the notations and terminology not defined here. There exists a neighbour multiset distinguishing 3-edge colouring of every graph G of minimum degree at least 103. If G is a regular graph, there exists a neighbour sum distinguishing 2-edge colouring of G if and only if it can be decomposed into 2 locally irregular subgraphs. To exemplify the fact that the two graph invariants representing the minimum numbers of colors necessary to create a neighbour multiset distinguishing edge colouring and a locally irregular edge colouring, resp., are distinct, let us consider a graph constructed as follows. It is easy to see that there exist multiset distinguishing 2-edge colourings of this graph, but no locally irregular 2-edge colouring This example may be generalized by substituting the paths of length 2 with any other even paths. The thesis follows by Theorem 8 (it is sufficient to choose a−v , a+v from these sets, resp., so that a−v , a+v ≡ t(v) (mod λv))
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