Abstract
We deal with a variant of the 1–2–3 Conjecture introduced by Gao, Wang, and Wu (Graphs Combin 32:1415–1421, 2016) . This variant asks whether all graphs can have their edges labelled with 1 and 2 so that when computing the sums of labels incident to the vertices, no monochromatic cycle appears. In the aforementioned seminal work, the authors mainly verified their conjecture for a few classes of graphs, namely graphs with maximum average degree at most 3 and series–parallel graphs, and observed that it also holds for simple classes of graphs (cycles, complete graphs, and complete bipartite graphs). In this work, we provide a deeper study of this conjecture, establishing strong connections with other, more or less distant notions of graph theory. While this conjecture connects quite naturally to other notions and problems surrounding the 1–2–3 Conjecture, it can also be expressed so that it relates to notions such as the vertex-arboricity of graphs. Exploiting such connections, we provide easy proofs that the conjecture holds for bipartite graphs and 2-degenerate graphs, thus generalising some of the results of Gao, Wang, and Wu. We also prove that the conjecture holds for graphs with maximum average degree less than \(\frac{10}{3}\), thereby strengthening another of their results. Notably, this also implies the conjecture holds for planar graphs with girth at least 5. All along the way, we also raise observations and results highlighting why the conjecture might be of greater interest.
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Notes
Throughout this work, we never refer to a cycle of length x as an x-cycle; every use of the term “x-cycle” always refers to a cycle in which all vertices have sum x by some (possibly partial) labelling.
Note that our goal is not to survey the whole field; we thus refer the interested reader to the references we provide for further information on the topic.
References
Ahadi, A., Dehghan, A., Sadeghi, M.-R.: Algorithmic complexity of proper labeling problems. Theor. Comput. Sci. 495, 25–36 (2013)
Bensmail, J., Hocquard, H., Marcille, P.-M.: Adding Direction Constraints to the 1-2-3 Conjecture. Preprint (2022). https://hal.archives-ouvertes.fr/hal-03900260
Bensmail, J., Li, B., Li, B.: An injective version of the 1-2-3 Conjecture. Graphs Combin. 37, 281–311 (2021)
Bensmail, J., Fioravantes, F., McInerney, F.: On the role of \(3\)s for the 1-2-3 Conjecture. Theor. Comput. Sci. 892, 238–257 (2021)
Bensmail, J., Fioravantes, F., Nisse, N.: On proper labellings of graphs with minimum label sum. Algorithmica 84, 1030–1063 (2022)
Bonamy, M.: Global discharging methods for coloring problems in graphs. Ph.D. thesis, Université de Montpellier, France (2015)
Bonamy, M., Bousquet, N., Hocquard, H.: Adjacent vertex-distinguishing edge coloring of graphs. In: Nešetřil, J., Pellegrini, M. (eds.) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol. 16. Edizioni della Normale, Pisa (2013)
Borodin, O.V., Kostochka, A.V., Nešetřil, J., Raspaud, A., Sopena, É.: On the maximum average degree and the oriented chromatic number of a graph. Discrete Math. 206, 77–89 (1999)
Chartrand, G., Kronk, H.V., Wall, C.E.: The point-arboricity of a graph. Isr. J. Math. 6, 169–175 (1968)
Dudek, A., Wajc, D.: On the complexity of vertex-coloring edge-weightings. Discrete Math. Theor. Comput. Sci. 13(3), 45–50 (2011)
Gao, Y., Wang, G., Wu, J.: A relaxed case on 1-2-3 Conjecture. Graphs Combin. 32, 1415–1421 (2016)
Hakimi, S.L., Schmeichel, E.F.: A note on the vertex arboricity of a graph. SIAM J. Discrete Math. 2, 64–67 (1989)
Kalkowski, M.: A note on the 1,2-Conjecture. Ph.D. thesis, Adam Mickiewicz University, Poland (2009)
Kalkowski, M., Karoński, M., Pfender, F.: Vertex-coloring edge-weightings: towards the 1-2-3 Conjecture. J. Combin. Theory Ser. B 100, 347–349 (2010)
Karoński, M., Łuczak, T., Thomason, A.: Edge weights and vertex colours. J. Combin. Theory Ser. B 91, 151–157 (2004)
Keusch, R.: A Solution to the 1-2-3 Conjecture. Preprint (2023). arXiv:2303.02611
Kronk, H.V., Mitchem, J.: Critical point arboritic graphs. J. Lond. Math. Soc. 9, 459–466 (1975)
Neumann-Lara, V.: The dichromatic number of a digraph. J. Combin. Theory Ser. B 33(3), 265–270 (1982)
Przybyło, J., Woźniak, M.: On a 1,2 Conjecture. Discrete Math. Theor. Comput. Sci. 12(1), 101–108 (2010)
Raspaud, A., Wang, W.: On the vertex-arboricity of planar graphs. Eur. J. Combin. 29, 1064–1075 (2008)
Seamone, B.: The 1-2-3 Conjecture and related problems: a survey. Preprint (2012). arXiv:1211.5122
Thomassen, C., Wu, Y., Zhang, C.-Q.: The \(3\)-flow conjecture, factors modulo \(k\), and the 1-2-3-conjecture. J. Combin. Theory Ser. B 121, 308–325 (2016)
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Bensmail, J., Hocquard, H. & Marcille, PM. On Inducing Degenerate Sums Through 2-Labellings. Graphs and Combinatorics 40, 23 (2024). https://doi.org/10.1007/s00373-024-02758-9
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DOI: https://doi.org/10.1007/s00373-024-02758-9