Further Evidence Towards the Multiplicative 1-2-3 Conjecture
aa r X i v : . [ c s . D M ] A p r Further Evidence Towards the Multiplicative 1-2-3 Conjecture
Julien Bensmail a , Hervé Hocquard b , Dimitri Lajou b , Éric Sopena b a Université Côte d’Azur, CNRS, Inria, I3S, France b Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400, Talence, France
Abstract
The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kaziów in 2012,states that, a few obvious exceptions apart, all graphs can be -edge-labelled so that notwo adjacent vertices get incident to the same product of labels. To date, this conjecturewas mainly verified for complete graphs and -colourable graphs. As a strong support tothe conjecture, it was also proved that all graphs admit such -labellings.In this work, we investigate how a recent proof of the multiset version of the 1-2-3Conjecture by Vučković can be adapted to prove results on the product version. We provethat -chromatic graphs verify the product version of the 1-2-3 Conjecture. We also provethat for all graphs we can design -labellings that almost have the desired property. Thisleads to a new problem, that we solve for some graph classes. Keywords: -chromatic graphs.
1. Introduction
This work takes place in the general context of distinguishing labellings , wherethe aim, given an undirected graph, is to label its edges so that its adjacent verticesget distinguished by some function computed from the labelling. Formally, a k -labelling ℓ : E ( G ) → { , . . . , k } of a graph G assigns a label from { , . . . , k } to each edge, and,for every vertex v , we can compute some function f ( v ) of the labels assigned to the edgesincident to v . The goal is then to design ℓ so that f ( u ) = f ( v ) for every edge uv of G .As reported in a survey [3] by Gallian on the topic, there actually exist dozens and dozenstypes of distinguishing labelling notions, which all have their own particular behavioursand subtleties.We are here more particularly interested in the so-called , which isdefined through the following notions. Given a labelling ℓ of a graph G , we can computefor every vertex v its sum σ ℓ ( v ) of incident labels, being formally σ ℓ ( v ) = Σ u ∈ N ( v ) ℓ ( uv ) .We say that ℓ is s-proper if the so-obtained σ ℓ yields a proper vertex-colouring of G , i.e., σ ℓ ( u ) = σ ℓ ( v ) for every edge uv . Generally speaking, not only we aim at finding s-proper k -labellings of G , but also we aim at designing such ones having k as small as possible.Thus, for G , we are interested in determining χ S ( G ) , which is the smallest k ≥ such thats-proper k -labellings of G do exist.Greedy arguments show that there exists only one connected graph G for which χ S ( G ) is not defined, and that graph is K . This implies that χ S ( G ) is defined for every graph G with no component isomorphic to K , which we call a nice graph . It is then legitimateto wonder how large can χ S ( G ) be in general, for a nice graph G . Karoński, Łuczak andThomason conjectured that this value cannot exceed in general [6]: If G is a nice graph, then χ S ( G ) ≤ . Preprint submitted to ... April 22, 2020 ne could naturally wonder about slight modifications of the 1-2-3 Conjecture, wherethe aim would be to design labellings ℓ distinguishing adjacent vertices accordingly to afunction f that is somewhat close to the sum function σ ℓ . There actually exist at leasttwo such variants, to be described in what follows, which sound particularly interestingdue to their respective subtleties, to some behaviours they share with the original 1-2-3Conjecture, and to general existing connections with that conjecture. • The first such variant we consider is the one where adjacent vertices of a graph G arerequired, by a labelling ℓ , to be distinguished by their multisets of incident labels.Recall that a multiset is a set in which elements can be repeated. For a vertex v of G , we denote by µ ℓ ( v ) the multiset of labels assigned to the edges incident to v . Wesay that ℓ is m-proper if µ ℓ is a proper vertex-colouring of G , while we denote by χ M ( G ) the least k ≥ such that G admits m-proper k -labellings (if any). • The second such variant is the one where adjacent vertices of G must be, by ℓ ,distinguished accordingly to the products of their incident labels. Formally, for avertex v of G , we define ρ ℓ ( v ) as the product of labels assigned to the edges incidentto v . We say that ℓ is p-proper if ρ ℓ is a proper vertex-colouring of G . We denote by χ P ( G ) the smallest k ≥ such that G admits p-proper k -labellings (if any).There exist several interesting connections between the previous three series of no-tions. For instance, it can be easily noted that an s-proper or p-proper labelling is al-ways m-proper. As a result, χ M ( G ) ≤ min { χ S ( G ) , χ P ( G ) } holds for every graph G forwhich the parameters are defined (see below). In general, there is no other systematicrelationship between these three notions, though some exist in particular contexts. Forinstance, s-proper -labellings, m-proper -labellings and p-proper -labellings are equiv-alent notions in regular graphs [2]. It can also be noted that s-proper { , } -labellingsand p-proper { , } -labellings are equivalent notions [10]. Another illustration is that anm-proper k -labelling yields a p-proper { l , . . . , l k } -labelling, for any set { l , . . . , l k } of k pairwise coprime integers.Just as for the 1-2-3 Conjecture, one can wonder how large can χ M ( G ) and χ P ( G ) befor a given graph G . Before providing hints on that very question, let us first mentionthat, similarly as for s-proper labellings, the only connected graph admitting no m-properlabellings and no p-proper labellings is K . Thus, the notion of nice graph coincides forthe three types of proper labellings. It actually turns out that the straight analogue ofthe 1-2-3 Conjecture is believed to hold for m-proper labellings and p-proper labellings;namely: If G is a nice graph, then χ M ( G ) ≤ . If G is a nice graph, then χ P ( G ) ≤ . The multiset version of the 1-2-3 Conjecture was introduced by Addario-Berry, Aldred,Dalal and Reed in [1], while the product version was introduced by Skowronek-Kaziówin [9]. By an argument above, recall that the sum version and the product version ofthe 1-2-3 Conjecture, if true, would actually imply the multiset version. From that angle,the multiset version does appear, at least intuitively, as the most feasible out of the threeversions. This is reinforced by unique behaviours of m-proper labellings over s-properlabellings and p-proper labellings. In particular, by a labelling ℓ of a graph, in order tohave µ ℓ ( u ) = µ ℓ ( v ) for any two vertices u and v , note that we must have d ( u ) = d ( v ) .The most notable facts towards these three variants of the 1-2-3 Conjecture to dateare: 2 Regarding the sum version, the best result to date, proved by Kalkowski, Karońskiand Pfender in [5], is that χ S ( G ) ≤ holds for every nice graph G . The conjecture wasverified for all -colourable graphs [6]. Regarding -chromatic graphs, the conjecturewas verified for -edge-connected ones [12]. In [7], it was recently shown that χ S ( G ) ≤ holds for every nice regular graph G . • Regarding the multiset version, for long the best result, proved by Addario-Berry,Aldred, Dalal and Reed in [1], was that χ M ( G ) ≤ holds for every nice graph G . Afew years ago, a breakthrough result was obtained by Vučković in [11], in which hegave a full proof of the conjecture. Theorem 1.1 ([11]) . If G is a nice graph, then χ M ( G ) ≤ . • Regarding the product version, the best results to date were mainly obtained viaadaptations of arguments used to provide results towards the sum and multiset ver-sions. Specifically, Skowronek-Kaziów proved in [9] that χ P ( G ) ≤ holds for all nicegraphs G . In the same article, she proved the product version of the 1-2-3 Conjecturefor -colourable graphs.In this work, we provide several results towards the product version of the 1-2-3 Con-jecture. A first (minor) reason for focusing on this version is that it is, out of the threeversions, the least investigated one to date. A second (major) reason stems from the recentproof of Theorem 1.1 by Vučković. As pointed out earlier, m-proper labellings and p-properlabellings tend to have alike behaviours, which gives us hope that the proof of Vučkovićmight be a step towards proving the product version of the problem.Let us support this perspective further. It is first important to mention that the labels , , by a -labelling are very special in terms of vertex products. Note in particular thatlabel has a unique behaviour, since assigning label to an edge uv by a labelling ℓ impactsneither ρ ℓ ( u ) nor ρ ℓ ( v ) . It is important, however, to emphasise that assigning label to uv is not similar to deleting uv from the graph, as, though ℓ ( uv ) does not contribute to ρ ℓ ( u ) and ρ ℓ ( v ) , it requires ρ ℓ ( u ) and ρ ℓ ( v ) to be different by a p-proper -labelling ℓ . Because and are coprime, this implies that, in order for ρ ℓ ( u ) = ρ ℓ ( v ) to hold, the decompositionof ρ ℓ ( u ) into prime factors must differ from that of ρ ℓ ( v ) . In other words, if we denote by d i ( w ) the i -degree of a vertex w by a labelling as the number of edges incident to w assignedlabel i , then, by a -labelling ℓ , ρ ℓ ( u ) = ρ ℓ ( v ) holds if and only if either d ( u ) = d ( v ) or d ( u ) = d ( v ) .That last property makes labels and by a -labelling ℓ very close in terms of vertexmultisets and vertex products, since also µ ℓ ( u ) = µ ℓ ( v ) holds as soon as d ( u ) = d ( v ) or d ( u ) = d ( v ) . Thus, the difference between m-proper -labellings and p-proper -labellings only lies in the behaviour of label : for the first objects, every edge uv labelled contributes to both µ ℓ ( u ) and µ ℓ ( v ) , while, for the second objects, every edge uv labelled contributes to none of ρ ℓ ( u ) and ρ ℓ ( v ) . For that reason, m-proper -labellings are not p-proper in general; however, there are contexts where this is the case, such as the followingmeaningful one: Observation 1.2.
Nice regular graphs verify the product version of the 1-2-3 Conjecture.Proof.
Let G be a nice ∆ -regular graph. By Theorem 1.1, there exists an m-proper -labelling ℓ of G . We claim it is also p-proper. Indeed, by arguments above, if ρ ℓ ( u ) = ρ ℓ ( v ) holds for some edge uv , then d ( u ) = d ( v ) and d ( u ) = d ( v ) . Since d ( u ) = d ( v ) = ∆ , thismeans also d ( u ) = d ( v ) holds. We then deduce that µ ℓ ( u ) = µ ℓ ( v ) holds, a contradiction.Thus, no two adjacent vertices of G have the same product of labels.3ur main intention in this paper is to investigate how the mechanisms in the proofof Theorem 1.1 can be used in the product setting. In Section 2, we first prove that theproduct version of the 1-2-3 Conjecture holds for -chromatic graphs, which goes beyondthe best known such result to date for the sum version, which is, as stated earlier, that -edge-connected -chromatic graphs verify the sum version of the 1-2-3 Conjecture [12].In Section 3, we give a result that is close to the product version of the conjecture, as wedescribe how to design, for any nice graph, -labellings that are very close to be p-proper.This leads us to raising a conjecture on almost p-proper -labellings in Section 4, thatmatches an existing weakening of the sum version of the 1-2-3 Conjecture from [4]. Wefinally verify our conjecture for several classes of graphs.
2. The Multiplicative 1-2-3 Conjecture for -chromatic graphs For a graph G , a proper k -vertex-colouring is a partition ( V , . . . , V k ) of V ( G ) intoindependent sets, and the chromatic number χ ( G ) of G is the smallest k such that thereexist proper k -vertex-colourings of G . Recall that G is k -chromatic if its chromatic numberis exactly k . Equivalently, G is k -chromatic if it admits proper k -vertex-colourings, but noproper k ′ -vertex-colourings with k ′ < k .In this section, we prove the following: Theorem 2.1. If G is a -chromatic graph, then χ P ( G ) ≤ .Proof. Let ( V , V , V , V ) be a proper -vertex-colouring of G . For any vertex v ∈ V i , an upward edge (resp. downward edge ) is an incident edge going to a vertex in some V j with j < i (resp. j > i ). Note that all vertices in V have no upward edges, while all vertices in V have no downward edges.Free to move vertices from part to part, we may assume that, for every vertex v inany V i with i ∈ { , , } , there is an upward edge going to each of V , . . . , V i − . Indeed, ifthere is a j < i such that v has no upward edge to V j , then by moving v to V j we obtainanother -vertex-colouring of G that is proper. By repeating this moving process as longas needed, we eventually reach a proper -vertex-colouring with the desired property. Inparticular, note that the process finishes since vertices are only moved to parts with lowerindex. Furthermore, since G is -chromatic, none of the four parts can become empty.A p-proper -labelling ℓ of G will be obtained through two main stages. The first stagewill consist in considering the vertices in V and V , and labelling their upward edges so thatparticular types of products are obtained for their vertices, guaranteeing that none of themare in conflict (i.e., have the same product of incident labels). In particular, these verticeswill be bichromatic , meaning that they have both -degree and -degree at least . On thecontrary, in general, a few cases apart, the vertices in V and V will be monochromatic ,meaning that they are not bichromatic. More precisely, for i ∈ { , , } , a vertex v is i -monochromatic if it is incident only to edges labelled i or and d i ( v ) > . In otherwords, a -monochromatic vertex is a vertex v with ρ ℓ ( v ) = 1 , while an i -monochromaticvertex with i ∈ { , } is a vertex v with ρ ℓ ( v ) = i x for some x ≥ . Right after the firststage, in a second stage, we will label the edges joining vertices in V and V to get rid ofall remaining conflicts.As mentioned in the introductory section, recall that two vertices can only be in conflictif they have the same -degree and the same -degree. This means that an i -monochromaticvertex and a j -monochromatic vertex can only be in conflict if i = j , and that a monochro-matic vertex and a bichromatic vertex can never be in conflict. More generally speaking,a vertex with -degree i and -degree j and a vertex with -degree i ′ and -degree j ′ canonly be in conflict if i = i ′ and j = j ′ . 4o ease the understanding of the proof, we start from ℓ being the labelling of G assigninglabel to all edges. We then modify some labels so that particular products are obtainedfor some vertices. We describe this modification process step by step, so that, after eachof these steps, we can point out the consequences of our modifications. In particular, thereader should keep in mind that, at any point, any existing conflict is intended to be dealtwith in later stages of the modification process. In, particular, note that, at the beginning,all vertices are -monochromatic and are thus all in conflict. Step 1: Labelling the upward edges of V and V . We first relabel all upward edges of the vertices in V . To that end, we consider every v ∈ V in arbitrary order, and apply the following:1. for every upward edge vu with u ∈ V , we set ℓ ( vu ) = 2 ;2. for every upward edge vu with u ∈ V , we set ℓ ( vu ) = 3 ;3. if currently v has odd -degree, then we pick an arbitrary upward edge vu with u ∈ V , and set ℓ ( vu ) = 3 .Recall that such upward edges exist by our original assumption on ( V , V , V , V ) . Also,note that, at this point, the vertices in V verify the following: Claim 2.2.
Every vertex of V is bichromatic with even -degree. Furthermore, at this point, every vertex v in V has all its downward edges (if any)assigned label by ℓ . Now, for every such v ∈ V , we modify the label of the upward edgesas follows:1. for every upward edge vu with u ∈ V , we set ℓ ( vu ) = 2 ;2. if currently v has even -degree, then we pick an arbitrary upward edge vu with u ∈ V , and set ℓ ( vu ) = 3 .Note that Claim 2.2 is not impacted by these modifications. Furthermore, it can bechecked that the vertices in V fulfil the following: Claim 2.3.
Every vertex v of V is bichromatic with odd -degree.Proof of the claim. If v has downward edges to V , then they are labelled in which case v is bichromatic (regardless of whether the second item applies or not). If v has no downwardedges, then the second item of the process applies, and v gets bichromatic by labelling an upward edge to V . In both cases, v has its -degree being of the desired parity. ⋄ Note that Claims 2.2 and 2.3 imply that any two adjacent vertices in V and V cannotbe in conflict, due to their different -degrees. Furthermore, it can be checked that, at thispoint, the vertices in V and V meet the following properties: Claim 2.4.
For every vertex of V : • all downward edges to V and V are labelled or ; • all upward edges to V are labelled . Claim 2.5.
For every vertex of V : • all downward edges to V and V are labelled ; all downward edges to V are labelled . All previous claims imply that, at the moment, only vertices in V and V can bein conflict. More precisely, every vertex of V is currently either -monochromatic or -monochromatic, while every vertex of V is either -monochromatic or -monochromatic.Thus, two adjacent vertices in V and V can only be in conflict if they are both -monochromatic. The next stage is dedicated to getting rid of these conflicts. Step 2: Labelling the edges between V and V . For every vertex v of G , we define its { , } -degree as the sum d ( v ) + d ( v ) of its -degree and -degree. It is important to mention that, in all modifications we applyto ℓ from this point on, the only way for the { , } -degree of a vertex v in V ∪ V tochange is via setting to the label of edges uv with u ∈ V ∪ V . In particular, note thatClaims 2.2 and 2.3 are not impacted by such modifications, as they only alter -degrees.Hence, through performing such modifications, adjacent vertices in V and V cannot getin conflict.For the whole step, we define H as the set of (connected) components induced bythe edges joining -monochromatic vertices of V and vertices of V (of any type). Notethat any two conflicting vertices at this point, i.e., adjacent vertices being currently -monochromatic, are part of a component H of H . In what follows, we call an H containingsuch a pair of conflicting vertices a conflicting component . Our main goal here is now toapply local label modifications to get rid of all conflicting components of H .It is important to note that two vertices from two distinct components H and H of H cannot be adjacent. Assume indeed that uv is an edge of G , where v ∈ V ( H ) ∩ V and u ∈ V ( H ) ∩ V . By definition of H , this means that v is -monochromatic, in which case H and H altogether induce a component of H . A consequence is that we can freely treatthe conflicting components of H independently.Consider a conflicting component H ∈ H . Note that we would be done with H if wecould get rid of all conflicts in H by relabelling its edges so that all vertices in H remain -monochromatic or -monochromatic, as, this way, no conflict with vertices in V or V would arise (by Claims 2.2 and 2.3). This is a configuration that can actually almost beattained, in the following sense: Claim 2.6.
For every vertex v in any part V i ∈ { V , V } of H , we can relabel the edgesof H with and so that d ( u ) is odd for every u ∈ V i \ { v } , and d ( u ) is even for every u ∈ V − i . Similarly, we can relabel the edges of H with and so that d ( u ) is even forevery u ∈ V i \ { v } , and d ( u ) is odd for every u ∈ V − i .Proof of the claim. Assume the conditions of the statement are not already met. So let usconsider any vertex u different from v that does not verify the desired condition. Since H is connected, there is a path P from u to v in H . Now traverse P from u to v , and, asgoing along, switch the label of every traversed edge from to and vice versa . Note thatthis switching procedure has the following effects: • for every internal vertex of P , the parity of its -degree is not altered; • for each of the two ends u and v of P , the parity of its -degree is altered.This way, note that u now satisfies its desired condition, while, for all vertices of H different from u and v , the situation regarding their desired condition has not changed.6y repeating this switching procedure as long as desired, we eventually get that allvertices different from v have their -degree meeting the desired parity condition. ⋄ To deal with H , we now apply certain label modifications depending on the surround-ings of H . We start off by considering the following three cases. In each case, it is implicitlyassumed that the previous ones do not apply. • Case 1.
There is a vertex v ∈ V ( H ) ∩ V with a neighbour w ∈ V ∪ V .Recall that v is -monochromatic (by definition of H ), and thus vw is currentlylabelled . According to Claim 2.6, in H we can relabel edges with to so that allvertices in V ( H ) ∩ V have even -degree while all vertices in V ( H ) ∩ V \ { v } haveodd -degree. If also v has odd -degree, then no conflict remains in H . Otherwise,i.e., v has even -degree, then we change the label of vw to . As a result, v now getsodd -degree as well, while w remains bichromatic with the same -degree. Thus, noconflict remains in H , and no new conflict is created in G . • Case 2.
There is a -monochromatic vertex u ∈ V ( H ) ∩ V with a -monochromaticneighbour v ∈ V .Since u is -monochromatic, by Claim 2.5 it has no neighbour in V ∪ V . Also, uv iscurrently labelled . As in the previous case, according to Claim 2.6 we can relabelwith and the edges of H to reach a situation where all vertices in V ( H ) ∩ V haveodd -degree while all vertices in V ( H ) ∩ V \ { u } have even -degree. If u also haseven -degree, then we are done. Otherwise, u has odd -degree (thus at least ).In that case, we assign label to uv . As a result, u gets bichromatic with no suchneighbour while v remains -monochromatic. Thus, no conflict remains in H , andno new conflict was created in G . • Case 3.
There is a -monochromatic vertex u ∈ V ( H ) ∩ V with p ≥ neighbours v , . . . , v p ∈ V ( H ) ∩ V .Because u is -monochromatic, by Claim 2.5 it has no neighbour in V ∪ V . Also,since previous Cases 1 and 2 did not apply, u has no -monochromatic neighbour in V , thus all v i ’s are -monochromatic, and the v i ’s have no neighbours in V ∪ V .We here consider H ′ = H − u . Let us denote by C , . . . , C r the components of H ′ .By Claim 2.6, in each C j ( ≤ j ≤ r ) we can relabel the edges with and sothat all vertices in V ( C j ) ∩ V have even -degree while all vertices in V ( C j ) ∩ V but maybe one of the v i ’s have odd -degree. Note that this can be attained sinceeach of the C j ’s contains at least one of the v i ’s. Finally, assign label to all uv i ’s.As a result, u becomes -monochromatic with -degree at least , while its only -monochromatic neighbours are possibly some of the v i ’s, in which case these have -degree . In C j , the only vertex (one of the v i ’s) that was possibly in conflictwith some vertices has turned bichromatic or -monochromatic, while its neighboursremain -monochromatic or -monochromatic. Recall also that the v i ’s that becamebichromatic cannot be adjacent to another bichromatic vertex, as, in particular, theyhave no neighbours in V ∪ V . Thus no conflict remains in H , and no new conflictwas created in G .Consider any remaining conflicting component H ∈ H . Because Cases 1 to 3 above didnot apply to H , the following holds: Claim 2.7.
For every remaining conflicting component H ∈ H : all -monochromatic vertices u ∈ V ( H ) ∩ V have degree in G ; • all -monochromatic vertices of H have no bichromatic neighbours in G .Proof of the claim. By definition of H , all vertices of V ( H ) ∩ V are -monochromatic,while all -monochromatic vertices of V ( H ) ∩ V have no neighbours in V ∪ V (Claim 2.5).Since H did not verify Case 1 above, it has no vertex in V ( H ) ∩ V with a neighbour in V ∪ V . Similarly, because Cases 2 and 3 above did not apply, H has no -monochromaticvertex in V ( H ) ∩ V having a -monochromatic neighbour in V or two -monochromaticneighbours in V ( H ) ∩ V . Then the claim holds. ⋄ Let us now repeatedly apply the following procedure to H : • As long as H has a -monochromatic vertex v ∈ V ( H ) ∩ V with two -monochromaticneighbours u , u ∈ V ( H ) ∩ V , we assign label to vu and vu .Note that this raises no conflict. On the one hand, u and u get -monochromaticwith -degree while v is their unique -monochromatic neighbour, and it has -degree .Recall that v is actually the unique -monochromatic neighbour of u and u , since u and u have degree in the whole of G . Conversely, u and u are the only -monochromaticneighbours of v , since the only -monochromatic vertices we create in V ( H ) ∩ V duringthis procedure have degree in G , and thus in H . Thus, no new conflict arises.Once the previous procedure has been repeated as long as possible for every remainingconflicting component H of H , note that the remaining conflicts involve disjoint edges uv where u ∈ V , v ∈ V , and u and v are -monochromatic. Furthermore, since previousCases 1 to 3 did not apply to H , we deduce that u has degree precisely in G , that v hasno other -monochromatic neighbour in H , and that v has no neighbour in V ∪ V . Since G is nice, we must have d ( v ) ≥ , which means that v has other neighbours in V ( H ) ∩ V .Any such neighbour u ′ ∈ V ( H ) ∩ V must be -monochromatic. Indeed, on the one hand,if u ′ is -monochromatic, then the repeated process above could have been applied oncemore to H . On the other hand, note that u ′ cannot be -monochromatic with vu ′ beinglabelled , as, in the process above, only degree- vertices of G , and thus of V ( H ) ∩ V ,get -monochromatic. Now, we assign label to u ′ v and label to vu . As a result, u ′ remains -monochromatic, v gets bichromatic, while u gets -monochromatic. Thenno new conflict arises, because, in particular, u ′ remains -monochromatic with no suchneighbours, v cannot have bichromatic neighbours by Claim 2.7, and u is only neighbouring v . Eventually, the -labelling ℓ has no remaining conflicts, and is thus p-proper.
3. Restricted product conflicts by -labellings In this section, we show how the proof of Theorem 2.1 can be generalized, to provethat all graphs admit -labellings ℓ that are “almost” p-proper. By that, we mean thatif there are product conflicts by ℓ , then the structures induced by the conflicting verticesare somewhat weak. This is with respect to the following notion. Let ℓ be a labelling ofa graph G . For any x ≥ , we denote by S x the set of vertices v of G with ρ ℓ ( v ) = x .Rephrased differently, the product version of the 1-2-3 Conjecture states that every nicegraph G admits a -labelling such that S x is an independent set for every x ≥ .In the next result, we prove that every graph admits a -labelling where all S x ’s areindependent, with the exception of perhaps S , which might induce independent edges.8 heorem 3.1. Every graph G admits a -labelling such that S induces a (possibly empty)matching and isolated vertices while all other S x ’s are independent sets.Proof. We may assume that G is connected. If G is K , then it suffices to assign label toits only edge. If G is -colourable, then G admits a p-proper -labelling (according to [9]),by which every S x is an independent set. The same conclusion holds if G is -chromatic,by Theorem 2.1. Thus, we may suppose that G is k -chromatic for some k ≥ . Let usthus consider ( V , . . . , V k ) a proper k -vertex-colouring of G , where k = χ ( G ) . By similararguments as in the proof of Theorem 2.1, we may assume that every vertex v ∈ V i with i > has upward edges to every part V , . . . , V i − .Just as in the proof of Theorem 2.1, we start from a labelling ℓ of G assigning label to all edges. We then consider the vertices of V k , V k − , . . . , V following that order andmodify the labels of their upward edges so that certain products are obtained. Eventually,we will handle the edges joining the vertices in V and V so that additional conditions aremet to make sure that only particular conflicts remain.During a first modification phase, we aim at having the vertices verifying the following: • v ∈ V : -monochromatic or -monochromatic; • v ∈ V : -monochromatic or -monochromatic; • v ∈ V : bichromatic, -degree , and even { , } -degree; • v ∈ V : bichromatic, -degree , and odd { , } -degree; • v ∈ V : bichromatic, -degree , and even { , } -degree; • ... • v ∈ V n , n ≥ : bichromatic, -degree n , and odd { , } -degree; • v ∈ V n +1 , n ≥ : bichromatic, -degree n , and even { , } -degree; • ...We note that if we can produce a -labelling with the vertex properties above, thenthe only possible conflicts would be along edges uv such that u ∈ V , v ∈ V , and both u and v are -monochromatic. Indeed, two vertices u and v such that u ∈ V ∪ V and v ∈ V ∪· · ·∪ V k cannot be in conflict since monochromatic vertices and bichromatic verticescannot be in conflict. Two vertices u and v with u ∈ V n and v ∈ V n ′ +1 for n ≥ and n ′ ≥ cannot be in conflict since vertices with different { , } -degrees cannot be in conflict.Finally, two vertices u and v with u ∈ V n + p and v ∈ V n ′ + p for n ≥ , n ′ ≥ ( n = n ′ ) and p ∈ { , } cannot be in conflict since bichromatic vertices can only be in conflict if theyhave the same -degree and -degree.Let us now describe how to modify ℓ so that the conditions above are met. We considerthe vertices of V k , . . . , V following that order, from bottom to top, and modify labelsassigned to upward edges. An important condition we will maintain, is that every vertexin an odd part V n +1 ( n ≥ ) has all its downward edges (if any) labelled or , whileevery vertex in an even part V n ( n ≥ ) has all its downward edges (if any) labelled or . Note that this is trivially verified for the vertices in V k , since they have no downwardedges.Assume we are currently considering a vertex v in, say, an even part V n with n ≥ .By the hypothesis above, all downward edges of v are labelled or . Since all upward9dges of v are currently labelled , the -degree of v is currently . Let us consider each ofthe n parts V , V , V , . . . , V n − . Recall that v has a neighbour u i in each of these parts.Then we modify the label of each such edge vu i so that it becomes . This way, note thatthe -degree of v becomes exactly n , as required. Note also that we do not spoil the desireddownward condition for the u i ’s. Now, depending on how many downward edges of v arelabelled , we claim that we can always turn to the label of one or two upward edgesso that v gets bichromatic with odd { , } -degree as desired. Indeed, if n ≥ , then wecan freely change to the label of an upward edge of v to each of V and V to get v asdesired. If n = 2 , then note that v might be missing at most one incident edge labelled .Indeed, if, on the one hand, an odd number of downward edges of v are labelled , then v is already bichromatic with odd { , } -degree. On the other hand, if an even number ofdownward edges of v are labelled , then v currently has even { , } -degree, in which casewe make it odd by changing to the label of an upward edge to V . This way, note that v has to become bichromatic.Similar arguments hold in the case when v lies in an odd part V n +1 with n ≥ . Again,all downward edges of v are labelled or , while all upward edges are currently labelled .We change to the label of an upward edge of v to each of the n parts V , V , V , . . . , V n sothat v has -degree n . Now, we can make sure that v is bichromatic with even { , } -degreein the following way. If n ≥ , then we can change to the label of an upward edge to V n and/or V , if needed. If n = 1 , then note that v currently has -degree . If an odd numberof downward edges are labelled , then v is already bichromatic with even { , } -degree.If an even number of downward edges are labelled , then we can change to the label ofan upward edge to V to achieve the same conclusion.By arguments above, only adjacent -monochromatic vertices in V and V can bein conflict. More precisely, recall that the vertices of V are -monochromatic or -monochromatic, while the vertices of V are -monochromatic or -monochromatic. An-other important property of the vertices in V , . . . , V k is that none of them has both -degree and odd { , } -degree at least (bichromatic). In particular, assuming that, lateron, we only turn vertices in V into this special type, no conflict can involve special vertices.As in the proof of Theorem 2.1, let us define H as the set of components induced bythe upward edges (all of which are currently labelled ) of the -monochromatic vertices of V . If H has no component on more than two vertices, then we are done. So let us focuson H ∈ H , a component with order at least . Here as well, no vertex of H is adjacent toa vertex in another component of H , so we can again freely deal with H without mindingthe other components. If no two -monochromatic vertices in H are adjacent, then we aredone with H . So let us assume some adjacent vertices of H are -monochromatic, and someof these -monochromatic vertices actually have at least two -monochromatic neighbours(as otherwise we would be done as well).We start by performing the following process:1. As long as H has a -monochromatic vertex v ∈ V ( H ) ∩ V with at least two -monochromatic neighbours u , u ∈ V ( H ) ∩ V , we do the following: • If v has a -monochromatic neighbour u ′ in V with d ( u ′ ) = 2 , then we set ℓ ( u ′ v ) = 3 . • Otherwise, we set ℓ ( vu ) = ℓ ( vu ) = 2 .By this process, every considered vertex v of V becomes either -monochromatic (firstcase), or -monochromatic with -degree (second case). In the first case, all neighbours of10 in V are special, -monochromatic or -monochromatic, thus not in conflict with v . In thesecond case, the neighbours of v in V are all special, -monochromatic or -monochromaticwith -degree different from , thus not in conflict with v .We go on with the following process:2. As long as H has a -monochromatic vertex u ∈ V ( H ) ∩ V with at least two -monochromatic neighbours v , v ∈ V ( H ) ∩ V , we do the following: • If u does not have a -monochromatic neighbour in V , then we set ℓ ( uv ) = ℓ ( uv ) = 3 . • Otherwise, we do nothing.Note that by repeatedly applying the first of these steps, no new conflict arises. Thisis because all -monochromatic vertices we create in V ( H ) ∩ V have -degree , whileall -monochromatic vertices we create in V ( H ) ∩ V have -degree while all their -monochromatic neighbours have -degree .Let us now have a look at the subgraph B of H induced by its remaining -monochromaticvertices. Let us more particularly focus on the components of B . If no such component hasorder greater than , then we are done. So let us focus on one component B with order atleast . Since previous Steps 1 and 2 have been performed as long as possible, B must bea star with center u ∈ V and at least two leaves v , v in V , and u has -monochromaticneighbours in V .Consider now the subgraph C of G obtained from the vertices in B by adding theincident edges to their -monochromatic neighbours in V . Note that this graph mighthave several components; let us focus on one of these components, say C . By construction,all vertices in V ( C ) ∩ V are -monochromatic. Also, C contains a -monochromatic vertex u ∈ V with two -monochromatic neighbours v , v ∈ V . Note furthermore that v and v have degree in C (as otherwise Step 1 above could have been applied once more). Wenow modify the labelling using the following analogue of Claim 2.6 (we omit a proof, as itwould go along the exact same lines): Claim 3.2.
For every vertex v in any part V i ∈ { V , V } of C , we can relabel the edgesof C with and so that d ( u ) is odd for every u ∈ V i \ { v } , and d ( u ) is even for every u ∈ V − i . Similarly, we can relabel the edges of C with and so that d ( u ) is even forevery u ∈ V i \ { v } , and d ( u ) is odd for every u ∈ V − i . We now use Claim 3.2 as follows: • If | V ( C ) ∩ V | is even, then, by Claim 3.2, we can relabel the edges of C with and so that all vertices in V ( C ) ∩ V have even -degree, while all vertices in V ( C ) ∩ V have odd -degree. • If | V ( C ) ∩ V | is odd, then the same conclusion can be achieved in the subgraph C − v , since | V ( C − v ) ∩ V | is even.By this modification, note that any two adjacent -monochromatic vertices of C havetheir -degrees being of distinct parity, and are thus not in conflict. Recall in particularthat, in earlier Step 2 above, all -monochromatic vertices created in V have even -degreeexactly while all -monochromatic vertices created in V have odd -degree exactly .Similarly, in earlier Step 1 above, every created -monochromatic vertex was created in V and has odd -degree exactly . Thus, no conflict can involve -monochromatic vertices.11inally, we note that, in the process above, the only possible remaining conflict isactually in the second case, along the edge uv since v remains -monochromatic while u might have -degree , and thus be -monochromatic as well.To finish off this section, let us mention that the labelling scheme developed in the proofof Theorem 3.1 has another implication for the product version of the ,raised by Przybyło and Woźniak in [8]. The 1-2 Conjecture asks whether every graphhas an s-proper -total-labelling ℓ , i.e., a -total-labelling (assigning labels to edges andvertices) so that σ tℓ ( u ) = σ tℓ ( v ) for every edge uv , where σ tℓ ( w ) = σ ℓ ( w ) + ℓ ( w ) for everyvertex w . In other words, in this type of labelling we are also allowed to locally alter vertexsums (via vertex labels) without spoiling neighbouring ones.In [9], Skowronek-Kaziów also introduced and studied the product version of the 1-2 Conjecture. By adapting existing proofs for the sum version of the 1-2 Conjecture,she mainly proved that every graph admits a p-proper total-labelling assigning labels in { , , } to the edges and labels in { , } to the vertices. By modifying the last stage ofour labelling scheme in the proof of Theorem 3.1, we get another proof of that result. Theorem 3.3.
Every graph G admits a p-proper total-labelling assigning labels in { , , } to the edges and labels in { , } to the vertices.Proof. Mimic the proof of Theorem 3.1, until the last stage, i.e., to the point wherevertices in V are -monochromatic or -monochromatic, while the vertices in V are -monochromatic or -monochromatic. Assign label to all vertices, so that the productsare not altered. To get a total-labelling as desired, we get rid of all remaining conflicts byjust making sure that all vertices of V become -monochromatic. To that end, we simplychange to the label of every vertex in V .
4. A conjecture for -labellings with restricted product conflicts From a more general perspective, according to Theorem 3.1, for every graph G we candesign a -labelling such that G [ S x ] is a forest for every x ≥ . One can naturally wonderwhether -labellings are powerful enough to achieve the same goal. As we did not manageto come up with any obvious reason why this could be wrong, we raise: Conjecture 4.1.
Every graph G can be -labelled so that G [ S x ] is a forest for every x ≥ . It is worth noting that Conjecture 4.1 matches a similar conjecture raised in [4] byGao, Wang and Wu in the sum context. They notably proved that the sum version ofConjecture 4.1 holds for graphs with maximum average degree at most and series-parallelgraphs. In what follows, as support, we prove Conjecture 4.1 (sometimes in an actuallystronger form) for three classes of graphs: complete graphs, bipartite graphs, and subcubicgraphs. Theorem 4.2.
Every complete graph K n admits a -labelling such that one of the S x ’sinduces an edge, while all other S x ’s are independent sets.Proof. We give an iterative labelling scheme which, starting from K , yields a desired -labelling for larger and larger complete graphs K n . To that end, we need a strongerhypothesis, namely that for every complete graph K n there is a desired -labelling withthe additional requirement that either there is no vertex incident only to edges labelled ,or there is no vertex incident only to edges labelled .12his is true for K : by assigning label to the only edge, we get a -labelling where S induces an edge (while there are no other S x ’s) and no vertex is incident only to edgeslabelled . Assume now our stronger claim is true for K n − for some n ≥ , and considera -labelling of K n − , with vertex set { v , . . . , v n − } , obtained by induction (thus with thedesired properties). Let us extend this labelling to the incident edges of a newly-addedvertex v n joined to all vertices in { v , . . . , v n − } , by assigning label to all edges incident to v n if no vertex in { v , . . . , v n − } is incident only to edges labelled , or by assigning label to all edges incident to v n if no vertex in { v , . . . , v n − } is incident only to edges labelled .Note that the -degree of all vertices in { v , . . . , v n − } grows by the same amount, either or . Thus, no new conflict involving two vertices in { v , . . . , v n − } arises. Now, regarding v n , its -degree is either the smallest possible ( ) or the largest possible ( n − ) for a vertexwith degree n − . By our choice of making v n incident to either only edges assigned label or only edges assigned label , we deduce that v n cannot be involved in a conflict. Thus,there remains only one conflict, and there is either no vertex in { v , . . . , v n } incident onlyto edges labelled , or no vertex in { v , . . . , v n } incident only to edges labelled . Thisconcludes the proof. Theorem 4.3.
Every connected bipartite graph G admits a -labelling such that one of the S x ’s induces at most one star and isolated vertices, while all other S x ’s are independentsets.Proof. Let v ∗ be any vertex of G . From v ∗ , we get a partition V ∪ · · · ∪ V d of V ( G ) whereeach V i contains the vertices at distance i from v ∗ . Note that V = { v ∗ } . Since G isbipartite, none of the V i ’s contains an edge. Furthermore, for every edge uv we have u ∈ V i and v ∈ V i +1 for some i . A part V i is said even if i is even, while V i is said odd otherwise.We produce a -labelling ℓ of G where every vertex in an even V i different from V haseven -degree, while every vertex in an odd V i has odd -degree. Note that the existenceof ℓ proves the claim, since, by such a labelling, v ∗ is the only vertex from an even V i thatcan be involved in conflicts. In particular, if one of the S x ’s induces a graph containing astar, then that star must be centered at v ∗ .We consider the vertices of G different from v ∗ successively, starting from those in thedeepest V i ’s, and, as going up, finishing with those in V . In the course of this process, letus consider v ∈ V i , a vertex in some V i all of whose incident edges going to V i +1 (if any)have been labelled. By definition of the V i ’s, there is at least one edge incident to v goingto V i − . We assign label to every such edge going to V i − , but maybe to one of them (towhich we instead assign label ) so that the -degree of v is of the desired parity.Once all vertices different from v ∗ have been treated that way, we end up with ℓ havingthe desired properties. Theorem 4.4.
Every subcubic graph G admits a -labelling such that all S x ’s induce aforest.Proof. We prove the claim by induction on | V ( G ) | + | E ( G ) | . As the claim can easily beproved when G is small, we focus on proving the general case, which we do by consideringthe possible cases for the minimum degree δ ( G ) of G . • First assume δ ( G ) = 1 , and let v be a degree- vertex of G . Let us consider G ′ = G − v .By the induction hypothesis, there is a -labelling of G ′ which is as desired. We extendthis labelling to G by assigning label to the edge incident to v . This way, note thatthe resulting labelling is as desired, since G ′ [ S ] gets added a pending or isolatedvertex. 13 Next assume δ ( G ) = 2 , and let u be a degree- vertex with neighbours v and w ofdegree at least . We here consider G ′ = G − u , which has a -labelling with thedesired properties. Let us first try to extend this labelling to G by assigning label to uv and uw . Note that if the desired properties are not met, then it must be because G ′ [ S ] has a path from v to w . In particular, both v and w have product , and eachof these two vertices is adjacent, in G ′ [ S ] , to another vertex. Then, assign label to uv and label to uw . Now the resulting labelling of G must be as desired, since thisremoved v from G ′ [ S ] , and added a pending or isolated path of length to G ′ [ S ] .This is because G is subcubic, which, at this point, implies that v has at most oneneighbour in S . • Lastly assume δ ( G ) = 3 , i.e., G is cubic, and consider u a degree- vertex withneighbours v, w, x of degree . We consider G ′ = G − u , which, again, has a -labelling with the desired properties. If we do not obtain a desired labelling of G when assigning label to uv , uw and ux , then it must be because, say, v and w have product and are joined by a path in G ′ [ S ] . By arguments above, due tothe bounded maximum degree of G , if we do not obtain a desired labelling whenassigning label to uv and label to uw and ux , then this must be because x hasproduct , and G ′ [ S ] contains a path from x to a neighbour of v . Then we deducethat, by the labelling of G ′ , in G ′ the two remaining neighbours of v have product and , and x has a neighbour with product . Then note that we are done whenassigning label to uv and ux , and label to uw . Indeed, this removes v from G ′ [ S ] and x from G ′ [ S ] , adds to G ′ [ S ] a pending or isolated edge (attached to v ), andadds to G ′ [ S ] a pending or isolated path of length (attached to u ).This concludes the proof. References [1] L. Addario-Berry, R.E.L. Aldred, K. Dalal, B.A. Reed. Vertex colouring edge parti-tions.
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