20.500.12556/DiRROS-20844 Bootstrap percolation in strong products of graphs Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider percolation numbers of strong products of graphs. If $G$ is the strong product $G_1\boxtimes \cdots \boxtimes G_k$ of $k$ connected graphs, we prove that $m(G,r)=r$ as soon as $r\le 2^{k-1}$ and $|V(G)|\ge r$. As a dichotomy, we present a family of strong products of $k$ connected graphs with the $(2^{k-1}+1)$-percolation number arbitrarily large. We refine these results for strong products of graphs in which at least two factors have at least three vertices. In addition, when all factors $G_i$ have at least three vertices we prove that $m(G_1 \boxtimes \dots \boxtimes G_k,r)\leq 3^{k-1} -k$ for all $r\leq 2^k-1$, and we again get a dichotomy, since there exist families of strong products of $k$ graphs such that their $2^{k}$-percolation numbers are arbitrarily large. While $m(G\boxtimes H,3)=3$ if both $G$ and $H$ have at least three vertices, we also characterize the strong prisms $G\boxtimes K_2$ for which this equality holds. Some of the results naturally extend to infinite graphs, and we briefly consider percolation numbers of strong products of two-way infinite paths. bootstrap percolation strong product of graphs infinite path ojačano pronicanje krepki produkt grafov neskončna pot true false true Angleški jezik Slovenski jezik Neznano 2024-11-20 09:36:29 2024-11-20 09:36:29 2024-11-21 03:36:18 0000-00-00 00:00:00 2024 0 0 22 str. iss. 4, article no. P4.35 Vol. 31 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2024-01-01 519.17:519.2 1077-8926 10.37236/11826 215693827 11826-PDF_file-52084-1-10-20241110.pdf 11826-PDF_file-52084-1-10-20241110.pdf 1 0221CDEFE7DA9103CF5B6F6C2C59DC4F 8e2ab2c3efe04af79731d18b64150b6ab0062a2e331f52da2b1743fe3bcfa123 c51524e9-a71a-11ef-ba3f-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=29439 https://www.combinatorics.org/ojs/index.php/eljc/article/view/v31i4p35 1 8d7e0262-a71a-11ef-ba3f-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=29438 Inštitut za matematiko, fiziko in mehaniko 0 0 0