Decomposability of graphs into subgraphs fulfilling the 1-2-3 Conjecture
aa r X i v : . [ m a t h . C O ] M a r Decomposability of graphs into subgraphs fulfilling the 1–2–3Conjecture
Julien Bensmail a,1 , Jakub Przyby lo b,2 a Universit´e Cˆote d’Azur, CNRS, I3S, Inria, France b AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30,30-059 Krakow, Poland
Abstract
The well-known 1–2–3 Conjecture asserts that the edges of every graph without isolatededges can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighteddegrees. This is open in general. We prove that every d -regular graph, d ≥
2, canbe decomposed into at most 2 subgraphs (without isolated edges) fulfilling the 1–2–3 Conjecture if d / ∈ { , , , , , } , and into at most 3 such subgraphs in theremaining cases. Additionally, we prove that in general every graph without isolatededges can be decomposed into at most 24 subgraphs fulfilling the 1–2–3 Conjecture,improving the previously best upper bound of 40. Both results are partly based onapplications of the Lov´asz Local Lemma. Keywords:
1. Introduction
A graph of order at least 2 cannot be irregular , i.e. its vertices cannot have pairwisedistinct degrees. This does not concern multigraphs though. The least k so that anirregular multigraph can be obtained from a given graph G by replacing each edge byat most k parallel edges is called the irregularity strength of G . This graph invariantwas introduced in [11], and investigated further in numerous papers as a particularmean for measuring the “level of irregularity” of graphs, see e.g. [3, 12, 13, 14, 15,18, 21, 23, 25, 26, 27]. Potential alternative definitions of “irregular graphs” were alsoinvestigated by Chartrand, Erd˝os and Oellermann in [10]. In [5] the authors introducedand initiated research devoted to so-called locally irregular graphs , i.e. graphs in whichadjacent vertices have distinct degrees. Already earlier a local version of irregularitystrength was studied in [20]. There Karo´nski, Luczak and Thomason considered the least k so that a locally irregular multigraph can be obtained from a given graph G = ( V, E )via, again, replacement of every edge with at most k parallel edges. This problem wasin fact originally formulated in terms of weightings, where by a k - edge-weighting of G The first author was supported by PEPS grant POCODIS. This work was partially supported by the Faculty of Applied Mathematics AGH UST statutorytasks within subsidy of Ministry of Science and Higher Education.
Preprint submitted to Elsevier November 9, 2018 e mean any mapping ω : E → { , , . . . , k } . For such w we may define the so-called weighted degree of or simply the sum at a given vertex v as: s ω ( v ) := X e ∈ E v ω ( e ) , where E v denotes the set of edges incident with v in G ; we shall usually write simply s ( v )instead of s ω ( v ) if this causes no ambiguities further on. So in this setting, the authorsof [20] were interested in the least k such that a k -edge-weighting ω of G exists so that s ω ( u ) = s ω ( v ) for every edge uv ∈ E – we say that u and v are sum-distinguished then(note we must assume that G contains no isolated edges to that end, i.e. that it has no K -components). They posed a very intriguing question, commonly known as the in the literature nowadays. Conjecture 1 (1–2–3 Conjecture).
For every graph G = ( V, E ) without isolated edgesthere exists a weighting ω : E → { , , } sum-distinguishing all neighbours in G . They confirmed it for 3-colourable graphs, i.e. for graphs with χ ( G ) ≤ Theorem 2 ([20]).
Every -colourable graph without isolated edges fulfills the 1–2–3Conjecture. This is also commonly known to hold in particular for complete graphs. In general theconjecture is however still widely open. The first constant upper bound, with 30 insteadof 3, was provided in [1], and then improved in [2] and [31]. The best general result thusfar was delivered by Kalkowski, Karo´nski and Pfender, who proved that it is sufficient touse weights 1 , , , ,
5, see [19]. This result was obtained via refinement and modificationof an algorithm developed by Kalkowski [17] (concerning a total analogue of the 1–2–3Conjecture, see e.g. [29]). Quite recently a complete characterization of bipartite graphsfor which it is sufficient to use just weights 1 and 2 was also provided by Thomassen, Wuand Zhang [30]. Note that graphs which require only one weight ( i.e.
1) are preciselythe locally irregular graphs.Another direction of research towards inducing local irregularity in a graph was de-veloped by Baudon et al. [5], this time via graph decompositions. In this paper, by a decomposition of a graph G we mean a partition of the set of its edges into subsets induc-ing subgraphs of G (usually with some specified features). We say a graph G = ( V, E )can be decomposed into k locally irregular subgraphs if E can be partitioned into k sets: E = E ∪ E ∪ . . . ∪ E k so that G i := ( V, E i ) is locally irregular (where we admit E i tobe empty) for i = 1 , , . . . , k . Equivalently, it means we may colour the edges of G withat most k colours so that each of these induces a locally irregular subgraph in G . In [5]it was conjectured that except for some family of exceptional graphs (each of which hasmaximum degree at most 3, see [5] for details), every connected graph can be decom-posed into 3 locally irregular subgraphs. This was then confirmed in [28] for graphs withsufficiently large minimum degree. Theorem 3 ([28]).
Every graph G with minimum degree δ ( G ) ≥ can be decom-posed into locally irregular subgraphs.
2n general it was also proved by Bensmail, Merker and Thomassen [9] that every con-nected graph which is not exceptional can be decomposed into (at most) 328 locally irreg-ular subgraphs, what was then pushed down to 220 such subgraphs by Luˇzar, Przyby loand Sot´ak [22]. See also [5, 6, 9, 22] for a number of partial and related results.Here we develop research initiated in [7], and related to the both concepts discussedabove. From now on we shall write a graph fufills the 1–2–3 Conjecture if there actuallyexists its neighbour sum-distinguishing 3-edge-weighting (assuming this holds in parti-cular for an edgeless graph). Though we are not yet able to prove the 1–2–3 Conjecture,even in the case of regular graphs, we shall prove below that for almost every d ≥
2, a d -regular graph G can be decomposed into 2 subgraphs fulfilling the 1–2–3 Conjecture ,while in the remaining cases it can be decomposed into 3 such subgraphs. At the endwe shall additionally prove that in general every graph without isolated edges can bedecomposed into (at most) 24 subgraphs consistent with the 1–2–3 Conjecture, whilethus far it was known that 40 such subgraphs were always sufficient, see [7] (also forother related results).
2. Basic Tools
We first present one basic observation followed by a recollection of a few fundamentaltools of the probabilistic method we shall use later on. For a vertex v of a given graph G = ( V, E ), by d S ( v ) we mean the number of edges uv ∈ E with u ∈ S if S ⊆ V , or thenumber of edges uv ∈ S in the case when S ⊆ E . Observation 4.
Every bipartite graph G can be decomposed into two subgraphs G and G such that for every vertex v of G , d G ( v ) ∈ (cid:20) d G ( v ) − , d G ( v ) + 12 (cid:21) . Proof.
If the set U of the vertices of odd degree in G is nonempty, then add a newvertex u and join it by a single edge with every vertex in U ; denote the obtained graph by G ′ . If U = ∅ , set G ′ = G and denote any vertex of G ′ as u . As the degrees of all verticesin G ′ are even, there exists an Eulerian tour in it. We then start at the vertex u andtraverse all edges of G ′ once along this Eulerian tour colouring them alternately red andblue. Then the red edges in G induce its subgraph G consistent with our requirements.This follows from the fact that if all degrees in the bipartite graph G are even, then ithas to have an even number of edges, and thus the thesis holds in particular for u . (cid:3) The following standard versions of the Lov´asz Local Lemma can be found e.g. in [4].
Theorem 5 (The Local Lemma; Symmetric Version).
Let A be a finite family ofevents in any probability space. Suppose that every event A ∈ A is mutually independentof a set of all the other events in A but at most D , and that Pr ( A ) ≤ p for each A ∈ A .If ep ( D + 1) ≤ , (1) then Pr ( T A ∈A A ) > . heorem 6 (The Local Lemma; General Case). Let A be a finite family of eventsin any probability space and let D = ( A , E ) be a directed graph such that every event A ∈ A is mutually independent of all the events { B : ( A, B ) / ∈ E } . Suppose that thereare real numbers x A ( A ∈ A ) such that for every A ∈ A , ≤ x A < and Pr ( A ) ≤ x A Y B ← A (1 − x B ) . (2) Then Pr ( T A ∈A A ) > . Here B ← A (or A → B ) means that there is an arc from A to B in D , the so-called dependency digraph . The Chernoff Bound below can be found e.g. in [16] (Th. 2.1, page26). Theorem 7 (Chernoff Bound).
For any ≤ t ≤ np , P r (BIN( n, p ) > np + t ) < e − t np and P r (BIN( n, p ) < np − t ) < e − t np where BIN( n, p ) is the sum of n independent Bernoulli variables, each equal to withprobability p and otherwise.
3. Main Result for Regular Graphs
In this section we shall prove that for almost all integers d ≥
2, every d -regular graphcan be decomposed into two graphs fulfilling the 1–2–3 Conjecture, while in the remainingfew cases – into three such graphs. The first subsection below is devoted to small valuesof d ; more generally we investigate in it graphs with upper-bounded chromatic number. In [7] it was proved the following result (note it follows also by Corollary 11 below).
Theorem 8 ([7]).
Every graph G without isolated edges and with χ ( G ) ≤ can bedecomposed into graphs fulfilling the 1–2–3 Conjecture. As complete graphs are known to fulfill the 1–2–3 Conjecture, by Brooks’ Theorem wethus obtain that every d -regular graph with 2 ≤ d ≤ Lemma 9.
If the edges of a graph G without isolated edges can be -coloured with redand blue so that the induced red subgraph R and blue subgraph B satisfy χ ( R ) ≤ r ≥ and χ ( B ) ≤ b ≥ , then we can also do it in such a way that neither B nor R containsan isolated edge. roof. Start from a 2-colouring of the edges of G with χ ( R ) ≤ r ≥ χ ( B ) ≤ b ≥ K -components (note that each isolatedtriangle of G shall be monochromatic then). We shall show that if the number of theseis still positive, then we may “get rid” of any given such component, without creating anew one (thus getting a contradiction, and hence proving the thesis):Assume uv forms such a monochromatic, say blue K -component. Observe that:(1’) u and v must belong to the same component of R , as otherwise one may recolour uv red;(2’) if e is a red edge adjacent with uv , say e = uw , then the size k of the (red)component of R − e including w equals 1 (hence, all red paths originating at u or v mustbe of length 2, i.e. there are in particular no isolated red edges adjacent with uv ), asotherwise, if there was any such edge e with k = 0, we could recolour uw blue (we wouldnot create any red K -component then, as we would not change colours on some existingred path joining u and v ), while in the remaining cases ( i.e. when k ≥ e adjacent with uv ) we could also recolour uw blue.By (1’) and (2’) there must be a red path uwv in G , and hence, by (2’), d ( u ) = 2 = d ( v ). Therefore, as all isolated triangles of G are monochromatic, d ( w ) ≥
3, and thus by(2’) all edges incident with w except for uw and vw are blue. We may however recolour uv red and uw blue then. (cid:3) Lemma 10.
For each positive integer k , every graph G = ( V, E ) without isolated edgesand with χ ( G ) ≤ k can be decomposed into k (some possibly empty) subgraphs G , G ,. . . , G k such that χ ( G i ) ≤ and G i contains no isolated edges for i = 1 , , . . . , k . Proof.
We prove the lemma by induction with respect to k . For k = 1 it triviallyholds, so let us assume that k ≥
2. Partition V into (possibly empty) independentsets V , V , . . . , V k . Colour red every edge uv ∈ E such that u ∈ V i and v ∈ V j with i ≡ j (mod 3), and colour blue the remaining edges of G . Let R and B be the red and,resp., blue subgraphs of G . Note that χ ( B ) ≤
3, as V r ∪ V r ∪ V r ∪ . . . ∪ V k − r formsan independent set in B for r = 0 , ,
2. On the other hand, χ ( R ) ≤ k − , as there are noedges in R between any two of the three red subgraphs G [ V r ∪ V r ∪ V r ∪ . . . ∪ V k − r ]with r = 0 , ,
2. Note that by Lemma 9, we may assume that neither B nor R contains anisolated edge. By the induction hypothesis, R can be then decomposed into subgraphs G , G , . . . , G k − with χ ( G i ) ≤ G k := B thus yields the thesis. (cid:3) By Lemma 10 and Theorem 2 we obtain the following corollaries.
Corollary 11.
Every graph G without isolated edges can be decomposed into ⌈ log χ ( G ) ⌉ graphs fulfilling the 1–2–3 Conjecture. Corollary 12.
Every d -regular graph G with ≤ d ≤ can be decomposed into subgraphs fulfilling the 1–2–3 Conjecture. To prove divisibility into 2 such subgraphs of d -regular graphs with larger d , wefirst prove in the next Subsection 3.2 that they admit a special vertex partition. InSubsection 3.3, we then discuss a peculiar sufficient condition for a graph to fulfill the1–2–3 Conjecture. 5 .2. Random Vertex Partition Lemma 13.
The vertices of every d -regular graph G with d ≥ , d = 15 , , can bepartitioned into sets V and V such that if d ≡ r mod 2 for some r ∈ { , } , then:(i) ∀ v ∈ V : d V ( v ) ≥ r ;(ii) ∀ v ∈ V : d V ( v ) ≥ ;(iii) ∀ v ∈ V : d V ( v ) ≥ r ;(iv) ∀ v ∈ V : d V ( v ) ≥ . Proof.
Assume G = ( V, E ) is a d -regular graph with d ≥ d = 15 ,
17. To everyvertex we randomly and independently assign 0 or 1 – each with probability 1 /
2, anddenote the 2-colouring obtained by c . Set V = c − (0) , V = c − (1).Assume first that d is even ( i.e. , r = 0). For any given vertex v ∈ V , denote by: • A ( v ) – the event that v ∈ V and d V ( v ) ≤ • A ( v ) – the event that v ∈ V and d V ( v ) ≤ • A ( v ) – the event that v ∈ V and d V ( v ) ≤ • A ( v ) – the event that v ∈ V and d V ( v ) ≤ c ,then the thesis shall be fulfilled. As drawings for all vertices are independent, for every v ∈ V we have: P r ( A ( v )) = P r ( v ∈ V ) · P r ( d V ( v ) ≤ P r ( v ∈ V ) · [ P r ( d V ( v ) = 0) + P r ( d V ( v ) = 1)]= 12 · "(cid:18) (cid:19) d + d (cid:18) (cid:19) d = (1 + d ) (cid:18) (cid:19) d +1 , (3)and analogously, for every i = 2 , , P r ( A i ( v )) = (1 + d ) (cid:18) (cid:19) d +1 . (4)Note now that every event A i ( v ) is mutually independent of all other events A j ( u ) with u at distance at least 3 from v , i.e. of all but at most 4 d + 3 other events. Therefore, by(1), (3) and (4), in order to apply the Lov´asz Local Lemma it is sufficient to show that: e (1 + d ) (cid:18) (cid:19) d +1 (4 d + 4) < d ≥
14. For d = 14 the left-hand side of inequality (5) takes value(approximately) 0 . ... <
1, while for d ≥
16 inequality (5) is implied by the followingone: 2 e (1 + d ) < d , √ e (1 + d ) < d , which holds as for f ( d ) := 2 d/ − √ e (1 + d ), we have f ′ ( d ) = 2 d/ / − √ e > d ≥
16 and f (16) ≈ . > P r \ v ∈ V ( A ( v ) ∩ A ( v ) ∩ A ( v ) ∩ A ( v )) ! > . The thesis follows.Assume now that d is odd ( i.e. , r = 1) and d ≥
19. In order to optimize our approachwe shall now have to aggregate the events concerning our requirements (i)–(iv) . Thusfor a vertex v ∈ V , denote the following (aggregated) event: • B ( v ): ( v ∈ V ∧ d V ( v ) ≥ ∧ d V ( v ) ≥ ∨ ( v ∈ V ∧ d V ( v ) ≥ ∧ d V ( v ) ≥ B ( v ) holds for every v ∈ V is positive.Note that for every v ∈ V , P r (cid:16) B ( v ) (cid:17) = P r (( v ∈ V ∨ d V ( v ) ≤ ∨ d V ( v ) ≤ ∧ ( v ∈ V ∨ d V ( v ) ≤ ∨ d V ( v ) ≤ P r (cid:16) B ( v ) | v ∈ V (cid:17) + 12 P r (cid:16) B ( v ) | v ∈ V (cid:17) = P r (cid:16) B ( v ) | v ∈ V (cid:17) = P r ( d V ( v ) ≤ ∨ d V ( v ) ≤ P r ( d V ( v ) = 2) + P r ( d V ( v ) = 1) + P r ( d V ( v ) = 0)+ P r ( d V ( v ) = 1) + P r ( d V ( v ) = 0)= (cid:18) d (cid:19) (cid:18) (cid:19) d + d (cid:18) (cid:19) d + (cid:18) (cid:19) d + d (cid:18) (cid:19) d + (cid:18) (cid:19) d = (cid:18) d ( d − d + 2 (cid:19) (cid:18) (cid:19) d < ( d + 2) − ( d +1) . (6)Again an event B ( v ) is mutually independent of all other events B ( u ) with u at distanceat least 3 from v , i.e. of all but at most d < ( d + 2) − e ( d + 2) − ( d +1) ( d + 2) < d ≥
19, as this is equivalent to the fact that g ( d ) := 2 ( d +1) / − √ e ( d + 2) > d ≥ g (19) ≈ . > g ′ ( d ) =2 ( d − / ln 2 − √ e > d ≥ P r \ v ∈ V B ( v ) ! > . (cid:3) .3. Family of Graphs Fulfilling the 1–2–3 Conjecture For a subset S of the set of vertices of a given graph G , G [ S ] shall denote the subgraphinduced by S in G , while by G ∪ G we shall mean the sum of two graphs G = ( V , E ), G = ( V , E ) understood as the pair ( V ∪ V , E ∪ E ). An independent set in a graph G = ( V, E ) is a subset I of V such that no edge of G has both ends in I . We call it maximal (or an independent dominating set ) if every vertex in V r I has a neighbourin I . In order to prove the existence of a specific family of graphs fulfilling the 1–2–3Conjecture we shall apply in the following lemma a certain refinement of Kalkowki’salgorithm from [17], exploiting for this aim the concept of maximal independent sets;see [8] for a corresponding application of a mixture of these two ingredients. Lemma 14.
If a graph G = ( V, E ) contains a maximal independent set I such that thereexists a constant α ≥ so that for R := V r I , (1 ◦ ) d ( v ) ≤ α for every v ∈ I and (2 ◦ ) d ( v ) ≥ α + d R ( v )+12 for every v ∈ R ,then G fulfills the 1–2–3 Conjecture. Proof.
By (2 ◦ ), d ( v ) ≥ v ∈ R , and thus there are no isolated edges in G .Note also that since I is a maximal independent set in G , then:(3 ◦ ) d I ( v ) ≥ v ∈ R ,so for every vertex v ∈ R we may fix an edge e v joining v with some vertex in I .We shall construct a 3-edge-weighting of G sum-distinguishing its neighbours. Ini-tially we label all edges in G by 2. These shall be modified gradually, and by ω ( e ) weshall always understand the current weight of an edge e in a given moment of our on-going relabelling algorithm specified below, and similarly, by s ( v ) we shall understandthe current sum at a vertex v in G .Let G R = G [ R ] be the graph induced by R in G . We analyse every of its componentsone by one, in any fixed order, and modify the labels of some of the edges incident(in G ) with at least one vertex of this component. Suppose H is the next componentto be analysed within the on-going algorithm, and arbitrarily order its vertices linearlyinto a sequence v , v , . . . , v n . We shall analyse one vertex in the sequence after anotherstarting from v , for which we perform no changes. Suppose thus we are about to analysea vertex v j with j ≥ H has more than one vertex) and denote by N − H ( v j ) the setof neighbours of v j in H which precede it in the fixed linear ordering – we call these the backward neighbours of v j . Similarly we define the set E − H ( v j ) of the backward edges of v j , i.e. these joining v j with its backward neighbours in H . We shall now modify (ifnecessary) weights of some edges, in order to obtain a sum at v j (in G ) which is distinctfrom the sums of all its backward neighbours – this sum of v shall then not change in thefurther part of the algorithm. To obtain our goal we shall be allowed to perform changesonly on the edges incident with its backward neighbours, namely for every backwardedge v k v j ∈ E − H ( v j ) of v j ( i.e. with k < j ) we shall be allowed to modify the labels of v k v j and e v k so that the sum at v k does not change; more specifically, if prior to thisstep j we had ω ( e v k ) = 2 (and ω ( v k v j ) = 2), then we may increase the label of v k v j by1 and decrease the label of e v k by 1 (or perform no changes on these two edges), while8f priory we had ω ( e v k ) = 1, we may decrease the label of v k v j by 1 and increase thelabel of e v k by 1 (observe that then ω ( e v k ) ∈ { , } and ω ( v k v j ) ∈ { , , } ). Note thatsuch admitted operations allow us to change the label of every backward edge of v j byexactly 1 (or do nothing with this label). Hence we have available at least | E − H ( v j ) | + 1distinct sums at v j via these operations. We choose one of these sums which is distinctfrom the current sums of all backward neighbours of v j and denote it by s ∗ (it exists, as | E − H ( v j ) | + 1 = | N − H ( v j ) | + 1), and we perform (some of the) admissible changes describedabove so that s ( v j ) = s ∗ afterwards. As these changes do not influence sums (in G ) ofthe other vertices of H , v j is now sum-distinguished from all its backward neighbours,and s ( v j ) ≥ · d I ( v j ) + 1 · d R ( v j ) = 2( d ( v j ) − d R ( v j )) + d R ( v j )= 2 (cid:18) d ( v j ) − d R ( v j )2 (cid:19) ≥ α + 1 (8)by (2 ◦ ). This shall not change as we guarantee that the sums of v , v , . . . , v j shall notbe modified in the further part of the construction. After step n , all neighbours in H arethus sum-distinguished (in G ), and we continue in the same manner with a consecutivecomponent of G R , if any is still left. Note that performing such changes concerning onecomponent of G R does not influence the sums in the other components, hence at theend of our construction all neighbours in G R are sum-distinguished (in G ). On the otherhand, as due to our algorithm every edge incident with a vertex in I has final weight 1or 2, by (1 ◦ ) we obtain that for v ∈ I , s ( v ) ≤ d ( v ) ≤ α. (9)Hence, by (8) and (9), every vertex in R is also sum-distinguished from each of itsneighbours in I . As I is an independent set, we thus obtain a desired 3-edge-weightingof G . (cid:3) Theorem 15.
Every d -regular graph G with d ≥ , d = 15 , , can be decomposed intotwo graphs fulfilling the 1–2–3 Conjecture. Proof.
Let G = ( V, E ) be a d -regular graph with d ≥ d = 15 ,
17, and d ≡ r mod 2for some r ∈ { , } . Let then V = V ∪ V be a vertex partition consistent with the thesisof Lemma 13. Denote G := G [ V ], G := G [ V ].Let H be the bipartite graph induced by the edges between V and V . Then δ ( H ) ≥ (ii) and (iv) from Lemma 13. By Observation 4 we next decompose H into twosubgraphs H and H such that d H ( v ) ∈ (cid:20) d H ( v ) − , d H ( v ) + 12 (cid:21) , (10)and thus also d H ( v ) ∈ (cid:20) d H ( v ) − , d H ( v ) + 12 (cid:21) (11)for every vertex v ∈ V . 9et G ′ = G ∪ H , G ′ = G ∪ H . Obviously G ′ and G ′ constitute a decomposition of G . In order to finish the proof it is thus sufficient to prove that they are both consistentwith the assumptions of Lemma 14. We show this to hold for G ′ , as the reasoning for G ′ is precisely symmetrical.For this aim note first that by the definition of H , the set I := V is an independentset in G ′ , and by (ii) from Lemma 13 and (10) above, it is also maximal. We shall nowshow that (1 ◦ ) and (2 ◦ ) from Lemma 14 hold for G ′ with R = V and α := d − − r .By (iii) from Lemma 13, for every vertex v ∈ I = V , d H ( v ) ≤ d − − r, and hence, by (10): d G ′ ( v ) = d H ( v ) ≤ (cid:24) d H ( v )2 (cid:25) ≤ (cid:24) d − − r (cid:25) = d − − r r . Consequently, (1 ◦ ) holds.On the other hand, by (10), for every v ∈ R = V : d G ′ ( v ) = d G ( v ) + d H ( v ) ≥ d V ( v ) + (cid:22) d − d V ( v )2 (cid:23) ≥ d V ( v ) + d − d V ( v ) − d + d V ( v ) − ≥ d + d V ( v ) − − r d − − r d V ( v ) + 12= α + d R ( v ) + 12 , and thus (2 ◦ ) holds. (cid:3)
4. General Upper Bound for All Graphs
We conclude by showing that every graph without isolated edges can be decomposedinto a certain number K of graphs fulfilling the 1–2–3 Conjecture, where K ≤
24. Wethereby improve the previously best upper bound K ≤
40 from [7]. We start fromproving a lemma on the existence of a subset of edges with certain properties in graphswith sufficiently large minimum degree.
Lemma 16. If G = ( V, E ) is a graph with minimum degree δ ≥ + 10 , then thereis a subset S ⊆ E such that ≤ d S ( v ) ≤ d ( v ) − for every vertex v ∈ V . Proof.
Let ∆ = ∆( G ). For every vertex v ∈ V choose arbitrarily a subset F v ⊆ E v ofcardinality δ . Now randomly and independently for every vertex v ∈ V choose one edgein F v – each with equal probability ( i.e. δ − ) – and denote it by e v . For every v ∈ V denote the event: • A ( v ): |{ u ∈ N G ( v ) : e u = uv }| < + 1.10uppose v is a vertex of degree d ; note that by the Chernoff Bound: P r ( A ( v )) ≤ P r (cid:18) BIN (cid:18) d, δ − δ (cid:19) < + 1 (cid:19) < e − ( δ − δ d − − ) δ − δ d ≤ e − ( δ − δ d − δ − δ d ( − − )) δ − δ d < e − · − d , (12)and set x v := e − − d for such v . In order to apply the general version of the LocalLemma we define a dependency digraph D by joining A ( v ) with an arc to every A ( u ) forwhich there exists w ∈ V such that uw, vw ∈ F w ; note that there are at most dδ suchevents A ( u ) for v . Then, since e − x < − x + 0 , x for x > f ( x ) := e − − x x isdecreasing for x ≥ , we have: x v Y A ( u ) ← A ( v ) (1 − x u ) ≥ e − − d (cid:16) − e − − δ (cid:17) δd > e − − d (cid:16) − e − − δ + 0 . e − · − δ (cid:17) δd > e − − d (cid:18) e − e − − δ (cid:19) δd > e − − d (cid:18) e − e − − (cid:19) d > e − · − d . (13)By (12), (13) and Theorem 6 we thus conclude that there is a choice of edges e u , u ∈ V ,so that for every v ∈ V , |{ u ∈ N G ( v ) : e u = uv }| ≥ + 1 . It is then sufficient to set S = { e u : u ∈ V } to obtain 1 ≤ d S ( v ) ≤ d ( v ) − for each v ∈ V , as desired. (cid:3) We are now ready to prove a lemma resembling one of the observations ( i.e.
Lemma 4.5)from [9] (used there as an ingredient in research concerning graph decompositions into agiven finite number of locally irregular subgraphs). Lemma 16 above shall enable us tooptimize the thesis of the aforementioned Lemma 17 below.
Lemma 17.
Every graph G = ( V, E ) without isolated edges can be decomposed into twographs H and F such that: H is either empty or has minimum degree δ ( H ) ≥ , and F contains no isolated edges and has degeneracy less than + 10 . Proof.
We shall first gradually remove some vertices from a given graph G . As long asthere is still some vertex v of degree less than 10 + 10 in what is left of it, we remove v from our contemporary graph. At the end of this process, we denote the leftover of G by H ′ and let F ′ be the subgraph of G induced by all its edges with at least one end outside V ( H ′ ). Note that F ′ has degeneracy less than 10 + 10 , while δ ( H ′ ) ≥ + 10 or11 ′ is empty then. If H ′ is empty, the thesis holds. Otherwise, by Lemma 16, there exists S ⊆ E ( H ′ ) such that for every vertex v ∈ V ( H ′ ),1 ≤ d S ( v ) ≤ d H ′ ( v ) − . (14)Note that for every isolated edge uv of F ′ , one of its ends must belong to V ( H ′ ) – wethen arbitrarily choose one edge from S incident with this end and add it to F ′ providedthat no other edge adjacent to uv was earlier added to F ′ . After repeating this procedurefor every such isolated edge we obtain a graph F of F ′ ; note that the degeneracy of F isstill less than 10 + 10 (as we may place the ends of the isolated edges of F ′ togetherwith the vertices in V ( F ) r V ( F ′ ) at the end of the ordering witnessing the degeneracyof F , since these vertices induce a forest in F ). At the same time, by (14), the remainingsubgraph of G , denoted by H (formed from H ′ by removing edges from S transferred to F ′ ), fulfills: δ ( H ) ≥ . (cid:3) Theorem 18.
Every graph G without isolated edges can be decomposed into subgraphsfulfilling the 1–2–3 Conjecture. Proof.
By Lemma 17, G can be decomposed into a graph H which is either empty orhas minimum degree δ ( H ) ≥ and a graph F of degeneracy less than 10 +10 whichcontains no isolated edges. By Theorem 3, H can be further decomposed into 3 locallyirregular subgraphs (which obviously fulfill the 1–2–3 Conjecture). On the other hand,as χ ( F ) ≤ + 10 < , by Lemma 10, F can be decomposed into 21 graphs whichare 3-colourable and contain no isolated edges, and thus fulfill the 1–2–3 Conjecture byTheorem 2. (cid:3)
5. Concluding Remarks
Note that by Theorems 8 and 15 we know that any d -regular graph without isolatededges can be decomposed into 2 subgraphs fulfilling the 1–2–3 Conjecture if only d / ∈{ , , , , , } . The remaining cases apparently need a separate special treatment,but either way, by Corollary 12 every d -regular graph, d ≥
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