Saturation in the Hypercube and Bootstrap Percolation

Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $\Theta(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and Computin

The min-max edge q-coloring problem

In this paper we introduce and study a new problem named \emph{min-max edge $q$-coloring} which is motivated by applications in wireless mesh networks. The input of the problem consists of an undirected graph and an integer $q$. The goal is to color the edges of the graph with as many colors as possible such that: (a) any vertex is incident to at most $q$ different colors, and (b) the maximum size of a color group (i.e. set of edges identically colored) is minimized. We show the following results: 1. Min-max edge $q$-coloring is NP-hard, for any $q \ge 2$. 2. A polynomial time exact algorithm for min-max edge $q$-coloring on trees. 3. Exact formulas of the optimal solution for cliques and almost tight bounds for bicliques and hypergraphs. 4. A non-trivial lower bound of the optimal solution with respect to the average degree of the graph. 5. An approximation algorithm for planar graphs.Comment: 16 pages, 5 figure

Graph Saturation in Multipartite Graphs

Let $G$ be a fixed graph and let ${\mathcal F}$ be a family of graphs. A subgraph $J$ of $G$ is ${\mathcal F}$-saturated if no member of ${\mathcal F}$ is a subgraph of $J$, but for any edge $e$ in $E(G)-E(J)$, some element of ${\mathcal F}$ is a subgraph of $J+e$. We let $\text{ex}({\mathcal F},G)$ and $\text{sat}({\mathcal F},G)$ denote the maximum and minimum size of an ${\mathcal F}$-saturated subgraph of $G$, respectively. If no element of ${\mathcal F}$ is a subgraph of $G$, then $\text{sat}({\mathcal F},G) = \text{ex}({\mathcal F}, G) = |E(G)|$. In this paper, for $k\ge 3$ and $n\ge 100$ we determine $\text{sat}(K_3,K_k^n)$, where $K_k^n$ is the complete balanced $k$-partite graph with partite sets of size $n$. We also give several families of constructions of $K_t$-saturated subgraphs of $K_k^n$ for $t\ge 4$. Our results and constructions provide an informative contrast to recent results on the edge-density version of $\text{ex}(K_t,K_k^n)$ from [A. Bondy, J. Shen, S. Thomass\'e, and C. Thomassen, Density conditions for triangles in multipartite graphs, Combinatorica 26 (2006), 121--131] and [F. Pfender, Complete subgraphs in multipartite graphs, Combinatorica 32 (2012), no. 4, 483--495].Comment: 16 pages, 4 figure