\documentclass[tikz]{standalone}

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

\standaloneconfig{margin=12pt}

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



%snake graph outline
\draw[thick] (0,0) to node[midway,below,scale=0.75,yshift=1pt]{$2$} (1,0) to node[midway,right,scale=0.75,xshift=-1pt]{$\mu$} (1,1) to (0,1) to node[midway,left,scale=0.75,xshift=1pt]{$2$}(0,0);
\draw[thick] (1,0) to  (2,0) to node[midway,right,scale=0.75,xshift=-1pt]{$2$} (2,1);
\draw[thick] (1,2) to node[midway,above,scale=0.75,yshift=-1pt]{$b$} (2,2) to node[midway,right,scale=0.75,xshift=-1pt]{$3$} (2,1) to node[midway,above,scale=0.75,yshift=-2pt]{$a$} (1,1) to  (1,2);
\draw[thick] (1,2) to node[midway,left,scale=0.75,xshift=1pt]{$2$} (1,3) to node[midway,above,scale=0.75,yshift=-1pt]{$\lambda$} (2,3) to (2,2);
\draw[thick] (1,3) to (1,4) to node[midway,above,scale=0.75,yshift=-1pt]{$2$} (2,4) to node[midway,right,scale=0.75,xshift=-1pt]{$b$} (2,3);

\end{tikzpicture}

\end{document}