On Proper Labellings of Graphs with Minimum Label Sum

Abstract

The 1-2-3 Conjecture states that every nice graph G (without component isomorphic to K2) admits a proper 3-labelling, i.e., a labelling of the edges with 1, 2, 3 such that no two adjacent vertices are incident to the same sum of labels. Another interpretation of this conjecture is that every nice graph G can be turned into a locally irregular multigraph M , i.e., with no two adjacent vertices of the same degree, by replacing each edge by at most three parallel edges. In other words, for every nice graph G, there should exist a locally irregular multigraph M with the same adjacencies and having few edges.We study proper labellings of graphs with the extra requirement that the sum of assigned labels must be as small as possible. That is, given a graph G, we are looking for a locally irregular multigraph M * with the fewest edges possible that can be obtained from G by replacing edges with parallel edges. This problem is quite different from the 1-2-3 Conjecture, as we prove that there is no k such that M * can always be obtained from G by replacing each edge with at most k parallel edges.We investigate several aspects of this problem. We prove that the problem of designing proper labellings with minimum label sum is N Phard in general, but solvable in polynomial time for graphs with bounded treewidth. We also conjecture that every nice connected graph G admits a proper labelling with label sum at most 3 2 |E(G)| + O(1), which we verify for several classes of graphs.

Further results on an equitable 1-2-3 Conjecture

“…• Regarding the initiator gadget I α of length α used in the construction of G, by Theorem 2. 3 we get that must assign label 1 to exactly 4α edges and label 2 to exactly 6α + 1 edges.…”

“…As an example, let us mention that for every nice complete graph K n , for which χ Σ (K n ) = 3, there is a proper 3-labelling assigning label 2 only once [2]. Also, for every bipartite graph G with χ Σ (G) = 3, there exist proper 3-labellings assigning label 3 at most twice [3]. It might be that, in general, using three labels might be too powerful, in the sense that two labels are "almost enough".…”

On the role of 3s for the 1-2-3 Conjecture

“…One of the main reasons why these presumptions are supposed to hold, is the fact that, in general, it seems that nice graphs admit 2-labellings that are almost proper, in the sense that they only need a few 3s to design proper 3-labellings. Note that, if this was true, then indeed the presumptions from [6] and [5] above would be likely to hold. It is also worth mentioning that this belief on the number of 3s is actually a long-standing one of the field, as, in a way, it lies behind the 1-2 Conjecture raised by Przybyło and Woźniak [16], which states that we should be able to build a proper 2-labelling of every graph if we are additionally allowed to locally alter every vertex colour a bit.…”

“…In particular, to better understand the connection between proper labellings and proper vertex-colourings, the authors of [1,6] studied proper labellings for which the resulting vertex-colouring c is required to be close to an optimal proper vertex-colouring (i.e., with the number of distinct resulting vertex colours being close to the chromatic number). It is also worth mentioning the work done in [5], where proper labellings minimising the sum of labels assigned to the edges were investigated.…”

“…Each of these previous investigations led to presumptions of independent interest. In particular, it is believed in [6], that every nice graph G should admit a proper labelling where the maximum vertex colour is at most 2∆(G) (recall that ∆(G) and δ(G) are used to denote the maximum and the minimum degree, respectively, of any vertex of G), while, from [5], it is believed that every nice graph G should admit a proper labelling where the sum of assigned labels is at most 2|E(G)|. One of the main reasons why these presumptions are supposed to hold, is the fact that, in general, it seems that nice graphs admit 2-labellings that are almost proper, in the sense that they only need a few 3s to design proper 3-labellings.…”