On decomposing graphs of large minimum degree into locally irregular subgraphs

A \emph{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 sufficiently large minimum degree, $\delta(G)\geq 10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to 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].


Introduction
All graphs considered are simple and finite. We follow [6] for the notations and terminology not defined here. A locally irregular graph is a graph whose every vertex has the degree distinct from the degrees of all of its neighbours. In other words, it is a graph in which the adjacent vertices have distinct degrees. Motivated by a few well known problems in edge colourings and labelings, we investigate a (non-evidently) related concept of decompositions of graphs into locally irregular subgraphs. More precisely, we say that a graph G = (V, E) 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, i.e., E = E 1 ∪ E 2 ∪ . . . ∪ E k with E i ∩ E j = ∅ for i = j and H i := (V, E i ) is locally irregular for i = 1, 2, . . . , k. Naturally, instead of decomposing the graph G, we may alterably paint its edges with k colours, say 1, 2, . . . , k, so that every colour class induces a locally irregular subgraph in G. Such colouring is called a locally irregular kedge colouring of G. Equivalently it is just an edge colouring such that if an edge uv ∈ E has colour i ∈ {1, 2, . . . , k} assigned under it, then the numbers of edges coloured with i incident with u and v must be distinct. As a weakening of such property, we may require u and v to to differ in frequencies of any of the colours from {1, 2, . . . , k}, not specifically the colour i (which is assigned to uv). In other words, we may wish the adjacent vertices to have distinct multisets of their incident colours under a colouring c : E → {1, 2, . . . , k}.
We call such c a neighbour multiset distinguishing k-edge colouring then. Such variant of edge colourings has in fact already been investigated by Addario-Berry et al. in [2]. They proved that 4 colours are always sufficient to construct such a colouring for every graph containing no isolated edges, and provided the following improvement.
Theorem 1.1. There exists a neighbour multiset distinguishing 3-edge colouring of every graph G of minimum degree δ ≥ 10 3 .
Their research was motivated by the so called 1-2-3 Conjecture due to Karoński, Luczak and Thomason [8], yet another concept introducing 'local irregularity' into a graph. Let c : E → {1, 2, . . . , k} be an edge colouring of G with positive integers. For every vertex v we denote by s c (v) := u∈N (v) c(uv) the sum of its incident colours and call it the weighted degree of v. We say that c is a neighbour sum distinguishing k-edge colouring of G if s c (u) = s c (v) for all adjacent vertices u, v in G. Equivalently, instead of assigning integers from {1, 2, . . . , k} to the edges, one might strive to multiply them the corresponding numbers of times in order to create a locally irregular multigraph of G, i.e., a multigraph in which the adjacent vertices have distinct degrees. In  Thus far it is known that a neighbour sum distinguishing 5-edge colouring exists for every graph without isolated edges, see [7]. In fact our interest in locally irregular graphs originated from 1-2-3 Conjecture via the following easy observation from [5]. It is worth noting in this context that asymptotically almost surely a random d-regular graph can be decomposed into 2 locally irregular subgraphs for every constant d > d 0 , where d 0 is some large number, see [5]. In the same paper the authors investigate a special family T of graphs of maximum degree (at most) 3 whose every member might be constructed from a triangle by repeatedly performed the following operation: choose a triangle with a vertex of degree 2 in our constructed graph and append to this vertex either a hanging path of even length or a hanging path of odd length with a triangle glued to its other end. They posed the following conjecture. Conjecture 1.4. Every connected graph G which does not belong to T and is not an odd length path nor an odd length cycle can be decomposed into 3 locally irregular subgraphs.
The graphs excluded in the conjecture above were also proven to be the only connected graphs which do not admit decompositions into any number of locally irregular subgraphs. This conjecture was verified in [5] for some classes of graph, e.g., complete graph, complete bipartite graphs, trees, cartesian products of graphs with the desired property (hypercubes for instance), and for regular graphs with large degrees.
The main result of this paper is the following strengthening of Theorem 1.1, which confirms Conjecture 1.4 for graphs of sufficiently large minimum degree. Theorem 1.5. Every graph G with minimum degree δ ≥ 10 10 can be decomposed into three locally irregular subgraphs.
Its proof combines a probabilistic approach with some known theorems on degree constrained 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 indeed distinct, let us consider a graph constructed as follows. Take a single edge, say uv, and append two hanging paths of length 2 to the vertex u and another two hanging paths of length 2 to the vertex v. It is easy to see that there exist multiset distinguishing 2-edge colourings of such graph, none of which is locally irregular. Creating the later colouring requires 3 colours. This example may also be easily generalized by substituting the paths of length 2 with any other even paths.
In the following, given two graphs , usually subgraphs of a host graph G, by H 1 ∪ H 2 we shall mean the graph (V 1 ∪ V 2 , E 1 ∪ E 2 ). Moreover, we shall write H 2 ⊂ H 1 if V 2 ⊂ V 1 and E 2 ⊂ E 1 , and in case of H 2 ⊂ H 1 , we shall also write H 1 − E(H 2 ) to denote the graph obtained from H 1 by removing the edges of H 2 . Given a subset E ′ of edges of a graph G = (V, E), the graph induced by E ′ shall be understood as G ′ := (V, E ′ ).

Tools
We shall use the Lovász Local Lemma and the Chernoff Bound, classical tools of the probabilistic method, see e.g. [4] and [9], respectively.
Here B ← A (or A → B) means that there is an arc from A to B in D, the so called dependency digraph. 3 Theorem 2.2 (Chernoff Bound). For any 0 ≤ t ≤ np: where BIN(n, p) is the sum of n independent variables, each equal to 1 with probability p and 0 otherwise.
For the deterministic part of our proof we shall in turn use a consequence of the following theorem from [1] (see also [2,3] for similar degree theorems and their applications).
Then there exists a spanning subgraph H of G such that for every v ∈ V : Proof. For every vertex v ∈ V we have: Since both sides of the inequality above are integers, then in fact: Analogously, Thus the sets of integers both contain all remainders modulo λ v . The thesis follows then by Theorem 2.3 (it is sufficient to choose a − v , a + v from these sets, resp., so that a − v , a + v ≡ t(v) (mod λ v )). 4 3. Proof of Theorem 1.5

Notions
Let G = (V, E) be a graph of minimum degree δ ≥ 10 10 . In the following by d(v) we shall mean the degree of a vertex v in G, and we shall write d(v) p for short instead of (d(v)) p . Let us denote the auxiliary 'optimizing' constant where β ≈ 6.2 (6.19 < β < 6.2). In order to apply Corollary 2.4, we shall also need two auxiliary vertex labelings, say c 1 and c 2 , with certain regular features. Thus for every vertex v let us first randomly and independently choose one value in {0, 1, . . . , 2 ⌈log β d(v)⌉ − 1}, each with equal probability, and denote it by c 1 (v). Then let us independently repeat our drawing, i.e., again for every v ∈ V randomly and independently we choose one value in {0, 1, . . . , 2 ⌈log β d(v)⌉ − 1}, each with equal probability, and denote it by c 2 (v).
By our construction below it shall be clear that every edge whose one end has the degree at least β times bigger than the other will be 'safe' from any potential conflicts between its end-vertices. Some of the remaining edges shall require extra attention though, and shall thus be called 'risky'. We distinguish three kinds of these, i.e., we say that an edge uv with 1/βd(v) < d(u) < βd(v) is: where, given integers b and k with b ∈ {1, . . . , k}, by writing |a| < b (mod k) we mean that a is an integer which is congruent to one of the following: −b + 1, −b + 2, . . . , b − 1 modulo k. Denote the sets of risky edges of types 1, 2 and 3 by R 1 , R 2 and R 3 , respectively. For each v ∈ V , let us also denote:

Probabilistic Lemma
Proof. For every v ∈ V , let X v , Y v , Z v , T v be the random variables of the cardinalities of the sets A(v), B(v), C(v), F (v), resp., and let .24 , respectively. Consider a vertex v ∈ V , and let u be any of its neighbours with d(u) ∈ (1/βd(v), βd(v)). Note that hence by the total probability: Finally, since the choices for c 1 and c 2 are independent, by (5) and (6), Consequently, since all choices are independent and 2/d(v) 0.38 ≤ 4/d(v) 0.38 , by (4) and the Chernoff Bound we obtain: 6 By the total probability we thus obtain that: Analogously, by (5) and (7), Finally, again by the Chernoff Bound and (8): and hence, by the total probability, (11) Note that for every vertex v ∈ V , the events A v , B v , C v and F v depend only on the random choices for v and its adjacent vertices u with 1/βd(v) < d(u) < βd(v). Thus each of these events (corresponding to the vertex v) is mutually independent of all events except (possibly) for these corresponding to the vertex v itself, those corresponding to the neighbours v ′ of v with 1/βd(v) < d(v ′ ) < βd(v) and those corresponding to the neighbours v ′′ of such v ′ for which 1/βd(v ′ ) < d(v ′′ ) < βd(v ′ ). In order to construct a dependency digraph D necessary to apply Theorem 2.1, from each of the events A v , B v , C v , F v , we draw arrows pointing at all other events corresponding to the vertices w (w = v or w = v ′ or w = v ′′ ) with the properties described above, for v ∈ V . For any event L corresponding to a vertex v of degree d in G (i.e., L = A v , L = B v , L = C v or L = F v ), we then set By our construction, for every such L, d + D (L) ≤ 3 + 4d + 4d(⌊βd⌋ − 1) = 3 + 4d⌊βd⌋, where d + D (L) is the out-degree of L in D. Moreover, if L → Q in D, then Q is an event corresponding to a vertex w with 1 β 2 d < d(w) < β 2 d.