Failed zero forcing numbers of Kneser graphs, Johnson graphs, and hypercubes

Abstract

For a graph $G=(V,E)$ and an assignment of black and white colors to its vertices, the zero forcing color-change rule operates as follows: if a vertex u and all of its neighbors except v are black, then v is forced to change its color to black. A proper subset S of V is called a zero forcing set if by initially assigning the black vertices to be the elements of S and repeatedly applying this rule to G, all vertices are eventually forced to change their colors to black. Otherwise, S is called a failed zero forcing set. The maximum size of a failed zero forcing set of G is called the failed zero forcing number of G and is denoted by F(G). In this paper, we study the failed zero forcing numbers of three graph families: Kneser graphs K(n, r), Johnson graphs J(n, r), and hypercube graphs $Q_n$ . Specifically, we prove that $F(K(n, r))=\left( {\begin{array}{c}n\\ r\end{array}}\right) -(r+2)$, for $2\le r \le \frac{n-1}{2}$, $F(J(n, r))=\left( {\begin{array}{c}n\\ r\end{array}}\right) -(r+1)$, for $1\le r \le \frac{n}{2}$, and $F(Q_n)=2^n - n$, for $n \ge 2$ .