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3 added another picture

I like the Cayley graph of the symmetric group $S_3$ for the presentation $P$ with generators $x,y$ and relations $r=x^3,s=y^2,t=xyxy$. This gives a $2$-complex $K(P)$ with one $0$-cell, two $1$-cells, labelled $x,y$, and three $2$-cells labelled $r,s,t$. The Cayley graph of $P$, i.e. the $1$-skeleton of the universal cover of $K(P)$, is shown above (part of Fig 10.6 of Topology and Groupoids), where the solid lines are mapped to $x$, and the broken lines to $y$. Also shown is a choice of maximal tree. You can also see the lifts of the $2$-cells, but they can't all really be drawn since the corresponding universal cover is a $6$-fold cover, and there are not enough $2$-cells in the above diagram. The dots show where a relation starts.

Note that in group theory, a so called Schreier transversal for a subgroup $H$ of a free group $F$ with generating set $X$ can be seen as a maximal tree in the covering graph of the $1$-complex $K$ determined by the generating set $X$ of $F$, the covering being determined by the subgroup $H$ of $F$.

Another nice exercise for students is to ask them to construct two $3$-fold covers of $S^1 \vee S^1$ such that one is regular and the other is not. For the convenience of readers here is picture of the two graphs, but with unlabelled arrows! It nicely shows how one graph has 3-fold symmetry and the other does not.

2 added anothe comment

I like the Cayley graph of the symmetric group $S_3$ for the presentation $P$ with generators $x,y$ and relations $r=x^3,s=y^2,t=xyxy$. This gives a $2$-complex $K(P)$ with one $0$-cell, two $1$-cells, labelled $x,y$, and three $2$-cells labelled $r,s,t$. The Cayley graph of $P$, i.e. the $1$-skeleton of the universal cover of $K(P)$, is shown above (part of Fig 10.6 of Topology and Groupoids), where the solid lines are mapped to $x$, and the broken lines to $y$. Also shown is a choice of maximal tree. You can also see the lifts of the $2$-cells, but they can't all really be drawn since the corresponding universal cover is a $6$-fold cover, and there are not enough $2$-cells in the above diagram. The dots show where a relation starts.

Note that in group theory, a so called Schreier transversal for a subgroup $H$ of a free group $F$ with generating set $X$ can be seen as a maximal tree in the covering graph of the $1$-complex $K$ determined by the generating set $X$ of $F$, the covering being determined by the subgroup $H$ of $F$.

Another nice exercise for students is to ask them to construct two $3$-fold covers of $S^1 \vee S^1$ such that one is regular and the other is not.

1

I like the Cayley graph of the symmetric group $S_3$ for the presentation $P$ with generators $x,y$ and relations $r=x^3,s=y^2,t=xyxy$. This gives a $2$-complex $K(P)$ with one $0$-cell, two $1$-cells, labelled $x,y$, and three $2$-cells labelled $r,s,t$. The Cayley graph of $P$, i.e. the $1$-skeleton of the universal cover of $K(P)$, is shown above (part of Fig 10.6 of Topology and Groupoids), where the solid lines are mapped to $x$, and the broken lines to $y$. Also shown is a choice of maximal tree. You can also see the lifts of the $2$-cells, but they can't all really be drawn since the corresponding universal cover is a $6$-fold cover, and there are not enough $2$-cells in the above diagram. The dots show where a relation starts.

Note that in group theory, a so called Schreier transversal for a subgroup $H$ of a free group $F$ with generating set $X$ can be seen as a maximal tree in the covering graph of the $1$-complex $K$ determined by the generating set $X$ of $F$, the covering being determined by the subgroup $H$ of $F$.