Abstract
This article is built upon three main ideas. First, for a class of monomial ideals, it is proved that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for an arbitrary graph G, we construct a monomial ideal \(M_G\) and show that the chromatic number of G is equal to the codimension of \(M_G\). Finally, for a class of graphs, we give a formula that computes the chromatic polynomial of G, evaluated at the chromatic number of G, in terms of the codimension and multiplicity of \(M_G\). In particular, the formula applies to all graphs satisfying the Erds–Faber–Lovász conjecture.
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Acknowledgements
After living in the USA for many years, my family and I had to return to our home country Argentina to comply with visa requirements. In the midst of much adversity, my parents in law prepared an old quiet farm for us to live, and my parents supported us financially. My dear wife Danisa, who has remained my closest friend through the years, typed this article, and our five children helped by doing their chores and schoolwork without complaint. This work was not supported by any grants, but it received great support from my loved ones, which I gratefully acknowledge.
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Alesandroni, G. Monomial invariants applied to graph coloring. J Algebr Comb 58, 95–112 (2023). https://doi.org/10.1007/s10801-023-01235-5
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DOI: https://doi.org/10.1007/s10801-023-01235-5