Klein cordial trees and odd cyclic cordial friendship graphs

For a graph $G$ and an abelian group $A$, a labeling of the vertices of $G$ induces a labeling of the edges via the sum of adjacent vertex labels. Hovey introduced the notion of an $A$-cordial vertex labeling when both the vertex and edge labels are as evenly distributed as possible. Much work has since been done with trees, hypertrees, paths, cycles, ladders, prisms, hypercubes, and bipartite graphs. In this paper we show that all trees are $\mathbb{Z}_2^2$-cordial except for $P_4$ and $P_5$. In addition, we give numerous results relating to $\mathbb{Z}_m$-cordiality of the friendship graph $F_n$. The most general result shows that when $m$ is an odd multiple of $3$, then $F_n$ is $\mathbb{Z}_m$-cordial for all $n$. We also give a general conjecture to determine when $F_n$ is $\mathbb{Z}_m$-cordial.