Abstract
We prove that the product version of the 1-2-3 Conjecture, raised by Skowronek-Kaziów in 2012, is true. Namely, for every connected graph with order at least 3, we prove that we can assign labels 1, 2, 3 to the edges in such a way that no two adjacent vertices are incident to the same product of labels.
This is a preview of subscription content, log in via an institution to check access.
Notes
Recall that a proper k-vertex-colouring of a graph G is a partition \((V_1, \dots , V_k\)) of V(G) where all \(V_i\)’s are independent. The chromatic number \(\chi (G)\) of G is the smallest \(k \ge 1\) such that proper k-vertex-colourings of G exist. We say that G is k-colourable if \(\chi (G) \le k\).
References
Addario-Berry, L., Aldred, R.E.L., Dalal, K., Reed, B.A.: Vertex colouring edge partitions. J. Comb. Theory Ser. B 94(2), 237–244 (2005)
Alon, N.: Combinatorial Nullstellensatz. Comb. Probab. Comput. 8, 7–29 (1999)
Anholcer, M.: Product irregularity strength of graphs. Discret. Math. 309(22), 6434–6439 (2009)
Bensmail, J., Hocquard, H., Lajou, D., Sopena, É.: Further evidence towards the multiplicative 1-2-3 conjecture. Discret. Appl. Math. 307, 135–144 (2022)
Bensmail, J., Hocquard, H., Lajou, D., Sopena, É.: On a list variant of the multiplicative 1-2-3 conjecture. Graphs Comb. 38(3), 88 (2022)
Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Comb. 1, 6 (1998)
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. Comb. Theory Ser. B 100, 347–349 (2010)
Kalkowski, M., Karoński, M., Pfender, F.: A new upper bound for the irregularity strength of graphs. SIAM J. Discret. Math. 25(3), 1319–1321 (2011)
Kalkowski, M., Karoński, M., Pfender, F.: The 1-2-3-conjecture for hypergraphs. J. Graph Theory 85(3), 706–715 (2017)
Karoński, M., Łuczak, T., Thomason, A.: Edge weights and vertex colours. J. Comb. Theory Ser. B 91, 151–157 (2004)
Przybyło, J.: The 1-2-3 conjecture almost holds for regular graphs. J. Comb. Theory Ser. B 147, 183–200 (2021)
Przybyło, J., Woźniak, M.: On a 1,2 conjecture. Discret. Math. Theor. Comput. Sci. 12(1), 101–108 (2010)
Skowronek-Kaziów, J.: Multiplicative vertex-colouring weightings of graphs. Inf. Process. Lett. 112(5), 191–194 (2012)
Vučković, B.: Multi-set neighbor distinguishing \(3\)-edge coloring. Discret. Math. 341, 820–824 (2018)
Acknowledgements
The authors are grateful to the three anonymous referees for their careful reading of a previous version of the current work, which allowed to improve the general quality and correctness not only of the main proof, but also of the whole paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bensmail, J., Hocquard, H., Lajou, D. et al. A Proof of the Multiplicative 1-2-3 Conjecture. Combinatorica 43, 37–55 (2023). https://doi.org/10.1007/s00493-023-00003-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-023-00003-0