OICC Press Mathematical Sciences 2251-7456 16 2 (June 2022) 2021 05 07 Perfect 2-colorings of the Johnson graph J(9, 4) 10.1007/s40096-021-00404-6 EN Mehdi Alaeiyan Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16844, IR Efat Alaeiyan Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16844, IR Journal Article 2021 05 07 Abstract A perfect 2-coloring of a graph Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma$$\end{document} with matrix M={mij}i,j=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=\{m_{ij}\}_{i, j=1, 2}$$\end{document} is a coloring of the vertices Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma$$\end{document} with colors called {1,2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1, 2\}$$\end{document} such that the number of vertices of color j adjacent to a fixed vertex of color i is equal to mij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{ij}$$\end{document} . We state the matrix M is the parameter matrix. Each class of an equitable partition is the vertices with the same color. In this article, we classify the parameter matrices of whole perfect 2-colorings of the Johnson graph J (9, 4).