A bipartite graph $G=(L,R;E)$ where $V(G)=L\cup R$, $|L|=p$, $|R|=q$ is called a $(p,q)$-tree if $|E(G)|=p+q−1$ and $G$ has no cycles. A bipartite graph $G=(L,R;E)$ is a subgraph of a bipartite graph $H=(L',R';E')$ if $L\subseteq L'$, $R\subseteq R'$ and $E\subseteq E'$. In this paper we present sufficient degree conditions for a bipartite graph to contain a $(p,q)$-tree.
Let $G=(L,R;E)$ be a bipartite graph with color classes $L$ and $R$ and edge set $E$. A set of two bijections $\{\varphi_1 , \varphi_2\}$, $\varphi_1 , \varphi_2 :L \cup R \to L \cup R$, is said to be a $3$-biplacement of $G$ if $\varphi_1(L)= \varphi_2(L) = L$ and $E \cap \varphi_1^*(E)=\emptyset$, $E \cap \varphi_2^*(E)=\emptyset$, $\varphi_1^*(E) \cap \varphi_2^*(E)=\emptyset$, where$\varphi_1^*$, $\varphi_2^*$ are the maps defined on $E$, induced by $\varphi_1$, $\varphi_2$, respectively. We prove that if $|L|=p$, $|R|=q$, $3 \leq p \leq q$, then every graph $G=(L,R;E)$ of size at most $p$ has a $3$-biplacement.
In this paper we consider 2-biplacement without fixed points of paths and $(p, q)$-bipartite graphs of small size. We give all $(p, q)$-bipartite graphs $G$ of size q for which the set $\mathcal{S}^{*}(G)$ of all 2-biplacements of $G$ without fixed points is empty.
