The 1–2–3 Conjecture almost holds for regular graphs

Przybyło, Jakub

doi:10.1016/j.jctb.2020.03.005

Cited by 26 publications

(12 citation statements)

References 23 publications

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“…It is this conjecture that "energized the study of the irregularity strength", as stated in [17], and settled foundations for entire discipline, as it provided an inspiration for many related papers, concepts and intriguing questions, see e.g. [2,1,3,8,9,11,10,12,13,14,17,18,21,22,23,24,25,28,27,29,30,32,31,40,35,38,36,37,39,43,44,7]. The conjecture remains open after more than three decades since it was formulated.…”

Section: Introductionmentioning

confidence: 99%

On the asymptotic confirmation of the Faudree-Lehel Conjecture for general graphs

Przybyło,

Wei

2021

Preprint

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Given a simple graph G, the irregularity strength of G, denoted by s(G), is the least positive integer k such that there is a weight assignment on edges f : E(G) → {1, 2, . . . , k} attributing distinct weighted degrees: f (v) := u:{u,v}∈E(G) f ({u, v}) to all vertices v ∈ V (G). It is straightforward that s(G) ≥ n/d for every d-regular graph G on n vertices with d > 1. In 1987, Faudree and Lehel conjectured in turn that there is an absolute constant c such that s(G) ≤ n/d + c for all such graphs. Even though the conjecture has remained open in almost all relevant cases, it is more generally believed that there exists a universal constant c such that s(G) ≤ n/δ+c for every graph G on n vertices with minimum degree δ ≥ 1 which does not contain an isolated edge. In this paper we confirm that the generalized Faudree-Lehel Conjecture holds for graphs with δ ≥ n β where β is any fixed constant larger than 0.8. Furthermore, we confirm that the conjecture holds in general asymptotically. That is we prove that for any ε ∈ (0, 0.25) there exist absolute constants c 1 , c 2 such that for all graphs G on n vertices with minimum degree δ ≥ 1 and without isolated edges, s(G)thus extending in various aspects and strengthening a recent result of Przyby lo, who showed that s(G) ≤ n d (1 + 1 ln /19 n ) = n d (1 + o(1)) for d-regular graphs with d ∈ [ln 1+ n, n/ ln n], and improving an earlier general upper bound: s(G) < 6 n δ + 6 of Kalkowski, Karoński and Pfender.

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Section: Introductionmentioning

confidence: 99%

On the asymptotic confirmation of the Faudree-Lehel Conjecture for general graphs

Przybyło,

Wei

2021

Preprint

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“…Recall that mE(G) ≤ χ Σ (G)|E(G)| and χ Σ (G) ≤ 5 (see [9]) hold for every nice graph G. Thus mE(G) ≤ 5|E(G)| holds for every nice graph G, and even mE(G) ≤ 4|E(G)| holds when G is regular [12]. Moreover, for every graph satisfying the 1-2-3 Conjecture, even mE(G) ≤ 3|E(G)| holds.…”

Section: General Boundsmentioning

confidence: 99%

“…Quite recently, a characterisation of nice bipartite graphs G with χ Σ (G) = 3 was provided in [13]. Moreover, χ Σ (G) ≤ 4 holds for every nice regular graph G [12] and χ Σ (T ) ≤ 2 holds for every nice tree T [5].…”

Section: Introductionmentioning

confidence: 99%

On Proper Labellings of Graphs with Minimum Label Sum

Bensmail

Fioravantes

Nisse

2020

Lecture Notes in Computer Science

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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.

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“…In particular, let us recall that χ Σ (G) ≤ 3 whenever χ(G) ≤ 3 (see [12]), which implies that mE(G) ≤ mE 3 (G) ≤ 3|E(G)| holds here. Recently, Przybyło proved in [15] that χ Σ (G) ≤ 4 whenever G is regular, which implies that mE(G) ≤ mE 4 (G) ≤ 4|E(G)| holds. More results of this sort can be found e.g.…”

Section: Boundsmentioning

confidence: 99%

On Proper Labellings of Graphs with Minimum Label Sum

2022

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The 1-2-3 Conjecture, raised by Karoński, Łuczak and Thomason, states that almost every graph G 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, that may be attributed to Chartrand et al., is that almost every 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 of its edges by at most three parallel edges. In other words, for almost every graph G there should be a locally irregular multigraph M with the same adjacencies and having a relatively small number of edges. The 1-2-3 Conjecture, if true, would indeed imply that there is such anIn this work, we study proper labellings of graphs with the extra requirement that the sum of assigned labels must be as small as possible. In other words, given a graph G, we are looking for a locally irregular multigraph M * with the smallest number of edges possible that can be obtained from G by multiplying edges. This problem is actually quite different from the 1-2-3 Conjecture, as we prove that there is no absolute constant 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, covering algorithmic and combinatorial aspects. In particular, we prove that the problem of designing proper labellings with minimum label sum is N P-hard in general, but solvable in polynomial time for graphs with bounded treewidth. We also conjecture that for all almost every graph G there should be a proper labelling with label sum at most 2|E(G)|, which we verify for several classes of graphs.

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The 1–2–3 Conjecture almost holds for regular graphs

Cited by 26 publications

References 23 publications

On the asymptotic confirmation of the Faudree-Lehel Conjecture for general graphs

On the asymptotic confirmation of the Faudree-Lehel Conjecture for general graphs

On Proper Labellings of Graphs with Minimum Label Sum

On Proper Labellings of Graphs with Minimum Label Sum

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