AUT Journal of Mathematics and Computing

AUT Journal of Mathematics and Computing

Waldschmidt constant of some classes of hypergraphs

Document Type : Original Article

Authors
Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Abstract
In this paper, we present a formula for the Waldschmidt constant of edge ideals of some classes of hypergraphs. We also show that if $G$ is a simple graph with binomial edge ideal $J_G$, then the Waldschmidt constant of $J_G$ is always 2. Furthermore, the symbolic defect of umbrella hypergraphs is presented.
Keywords
Subjects

[1]     A. Alilooee and A. Banerjee, Symbolic powers of multipartite hypergraphs and Waldschmidt constant, arXiv preprint, (2021).
[2]     C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Seceleanu,
A. Van Tuyl, and T. Vu, The Waldschmidt constant for squarefree monomial ideals, J. Algebraic Combin., 44 (2016), pp. 875–904.
[3]     C. Bocci and B. Franci, Waldschmidt constants for Stanley-Reisner ideals of a class of simplicial complexes, J. Algebra Appl., 15 (2016), pp. 1650137, 13.
[4]     E. Camps Moreno, C. Kohne, E. Sarmiento, and A. Van Tuyl, On the Waldschmidt constant of square-free principal Borel ideals, Proc. Amer. Math. Soc., 150 (2022), pp. 4145–4157.
[5]     B. Chakraborty and M. Mandal, Invariants of the symbolic powers of edge ideals, J. Algebra Appl., 19 (2020), pp. 2050184, 19.
[6]     S. M. Cooper, R. J. D. Embree, H. T. Ha, and A. H. Hoefel` , Symbolic powers of monomial ideals, Proc. Edinb. Math. Soc. (2), 60 (2017), pp. 39–55.
[7]     V. Ene and J. Herzog, On the symbolic powers of binomial edge ideals, in Combinatorial structures in algebra and geometry, vol. 331 of Springer Proc. Math. Stat., Springer, Cham, [2020] ©2020, pp. 43–50.
[8]     F. Galetto, A. V. Geramita, Y.-S. Shin, and A. Van Tuyl, The symbolic defect of an ideal, J. Pure Appl. Algebra, 223 (2019), pp. 2709–2731.
[9]     D. R. Gaur and K. Makino, On the fractional chromatic number of monotone self-dual Boolean functions, Discrete Math., 309 (2009), pp. 867–877.
[10]  Y. Gu, H. T. Ha, J. L. O’Rourke, and J. W. Skelton` , Symbolic powers of edge ideals of graphs, Comm. Algebra, 48 (2020), pp. 3743–3760.
[11]  J. Herzog, T. Hibi, F. Hreinsdottir, T. Kahle, and J. Rauh´ , Binomial edge ideals and conditional independence statements, Adv. in Appl. Math., 45 (2010), pp. 317–333.
[12]  M. Janssen, T. Kamp, and J. Vander Woude, Comparing powers of edge ideals, J. Algebra Appl., 18 (2019), pp. 1950184, 19.
[13]  A. V. Jayanthan, A. Kumar, and V. Mukundan, On the resurgence and asymptotic resurgence of homogeneous ideals, Math. Z., 302 (2022), pp. 2407–2434.
[14]  P. Mantero and V. Nguyen, The structure of symbolic powers of matroids, Trans. Amer. Math. Soc., 379 (2026), pp. 3855–3894.
[15]  P. Mihok, Z. Tuza, and M. Voigt´             , Fractional P-colourings and P-choice-ratio, Tatra Mt. Math. Publ., 18 (1999), pp. 69–77.
[16]  B. Oltsik, Symbolic defect of monomial ideals, Comm. Algebra, 52 (2024), pp. 3996–4012.
[17]  A. Simis, W. V. Vasconcelos, and R. H. Villarreal, On the ideal theory of graphs, J. Algebra, 167 (1994), pp. 389–416.
[18]  T. Szemberg and J. Szpond, Waldschmidt constants for Stanley-Reisner ideals of a class of graphs, in Multigraded algebra and applications, vol. 238 of Springer Proc. Math. Stat., Springer, Cham, 2018, pp. 159– 167.