The Characterization of Graphs Whose Sandpile Group has Fixed Number of Generators

Let \(\mathcal{K}_k\) be the family of connected graphs G whose sandpile groups have minimal number of generators equal to \( | G |-k-1\), where \( | G | \) is the number of vertices of \(G\). We survey previous result on the characterization of \(\mathcal{K}_1\) and \(\mathcal{K}_2\), including complete characterizations for both cases. Furthermore, we shed some light on the characterization of \(\mathcal{K}_3\). Particularly, we give a minimal list of graphs that are forbidden for the regular graphs in \(\mathcal{K}_3\) and we use them to give a characterization of the regular graphs in \(\mathcal{K}_3\).

Recommended citation: Carlos A. Alfaro, Michael D. Barrus, John Sinkovic and Ralihe R. Villagrán. (2021) "The Characterization of Graphs Whose Sandpile Group has Fixed Number of Generators" Extended Abstracts EuroComb 2021. Trends in Mathematics, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83823-2_91