\documentclass[tikz]{standalone}

\usetikzlibrary{decorations.pathreplacing,decorations.markings}
\usetikzlibrary{patterns}

\begin{document}

\begin{tikzpicture}
%diagonals + diagonal labels
%\draw[gray,thick,dashed] (0,1) to %node[midway,right,yshift=2pt,xshift=-2pt]{ $1$} (1,0);
%\draw[gray,thick,dashed] (1,1) to node[midway,right,yshift=2pt,xshift=-2pt]{ $1$} (2,0);

%diagonals + diagonal labels
\draw[gray,thick,dashed] (0,1) to node[midway,right,yshift=2pt,xshift=-2pt]{$1$} (1,0);
\draw[gray,thick,dashed] (0,2) to node[midway,right,yshift=2pt,xshift=-2pt]{$1$} (1,1);
\draw[gray,thick,dashed] (1,2) to node[midway,right,yshift=2pt,xshift=-2pt]{$2$} (2,1);
\draw[gray,thick,dashed] (2,2) to node[midway,right,yshift=2pt,xshift=-2pt]{$2$} (3,1);
\draw[gray,thick,dashed] (2,3) to node[midway,right,yshift=2pt,xshift=-2pt]{$1$} (3,2);
\draw[gray,thick,dashed] (3,3) to node[midway,right,yshift=2pt,xshift=-2pt]{$1$} (4,2);

%snake graph outline
\draw[thick,green] (0,0) to node[midway,below,yshift=2pt]{\color{black} $a$} (1,0);
\draw[thick] (0,0) to node[midway,left,xshift=2pt]{$x_2$} (0,1);
\draw[thick] (1,0) to node[midway,right,xshift=-2pt]{$x_1$} (1,1);
\draw[thick,green] (1,1) to node[midway,above,yshift=-3pt]{\color{black} $\mu x_1$} (0,1);
\draw[thick] (0,1) to node[midway,left,xshift=2pt]{$x_1$} (0,2);
\draw[thick,green] (0,2) to node[midway,above,yshift=-2pt]{\color{black} $x_2$} (1,2);
\draw[thick] (1,2) to node[midway,right,xshift=-3pt]{$a$} (1,1);
\draw[thick] (1,2) to node[midway,above,yshift=-2pt]{$x_2$}(2,2);
\draw[thick,green] (2,2) to node[midway,right,xshift=-3pt,yshift=-7pt]{\color{black} $\lambda x_2$} (2,1);
\draw[thick] (2,1) to node[midway,below,yshift=2pt]{$x_1$}(1,1);
\draw[thick] (2,1) to node[below,yshift=2pt,midway]{$x_2$} (3,1);
\draw[thick,green] (3,1) to node[midway,right,xshift=-3pt]{\color{black} $x_1$} (3,2);
\draw[thick] (3,2) to node[midway,above,yshift=-2pt]{$a$} (2,2);
\draw[thick] (3,2) to node[midway,right,yshift=-8pt,xshift=-3pt]{$\mu x_1$} (3,3);
\draw[thick,green] (3,3) to node[midway,above,yshift=-2pt]{\color{black} $x_1$} (2,3);
\draw[thick] (2,3) to node[midway,left,xshift=2pt]{$x_2$} (2,2);
\draw[thick] (3,3) to node[midway,above,yshift=-2pt]{$x_2$} (4,3);
\draw[thick,green] (4,3) to node[midway,right,xshift=-3pt]{\color{black} $a$} (4,2);
\draw[thick] (4,2) to node[midway,below,yshift=2pt]{$x_1$} (3,2);
\end{tikzpicture}

\end{document}