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

For a graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V,E)$$\end{document} 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 Qn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_n$$\end{document} . Specifically, we prove that F(K(n,r))=nr-(r+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(K(n, r))=\left( {\begin{array}{c}n\\ r\end{array}}\right) -(r+2)$$\end{document}, for 2 <= r <= n-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le r \le \frac{n-1}{2}$$\end{document}, F(J(n,r))=nr-(r+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(J(n, r))=\left( {\begin{array}{c}n\\ r\end{array}}\right) -(r+1)$$\end{document}, for 1 <= r <= n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le r \le \frac{n}{2}$$\end{document}, and F(Qn)=2n-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(Q_n)=2<^>n - n$$\end{document}, for n >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document} .