2-biplacement without fixed points of (p,q)-bipartite graphs

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

3-biplacement of bipartite graphs

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

Bipartite embedding of (p,q)-trees

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

2-placement of (p,q)-trees

Let G = (L,R;E) be a bipartite graph such that V(G) = L∪R, |L| = p and |R| = q. G is called (p,q)-tree if G is connected and |E(G)| = p+q-1. Let G = (L,R;E) and H = (L',R';E') be two (p,q)-tree. A bijection f:L ∪ R → L' ∪ R' is said to be a biplacement of G and H if f(L) = L' and f(x)f(y) ∉ E' for every edge xy of G. A biplacement of G and its copy is called 2-placement of G. A bipartite graph G is 2-placeable if G has a 2-placement. In this paper we give all (p,q)-trees which are not 2-placeable

Placing bipartite graphs of small size II

In this paper we give all pairs of non mutually placeable (p,q)-bipartite graphs G and H such that 2 ≤ p ≤ q, e(H) ≤ p and e(G)+e(H) ≤ 2p+q-1

2-biplacement without fixed points of (p,q)-bipartite graphs

Tyt. z nagł.Abstract. 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 S*(G) of all 2-biplacements of G without fixed points is empty. Keywords: bipartite graph, packing, embedding

Bipartite embedding of (p, q)-trees

Tyt. z nagł.References p. 124.Dostępny również w formie drukowanej.ABSTRACT: Formulas. In this paper we present sufficient degree conditions for a bipartite graph to contain a (p, q)-tree. KEYWORDS: Bipartite graph, tree, embedding graph

Independent cycles and paths in bipartite balanced graphs

Bipartite graphs G = (L,R;E) and H = (L',R';E') are bi-placeabe if there is a bijection f:L∪R→ L'∪R' such that f(L) = L' and f(u)f(v) ∉ E' for every edge uv ∈ E. We prove that if G and H are two bipartite balanced graphs of order |G| = |H| = 2p ≥ 4 such that the sizes of G and H satisfy ||G|| ≤ 2p-3 and ||H|| ≤ 2p-2, and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs. This result implies easily that for integers p and k₁,k₂,...,kₗ such that $k_i ≥ 2$ for i = 1,...,l and k₁ +...+ kₗ ≤ p-1 every bipartite balanced graph G of order 2p and size at least p²-2p+3 contains mutually vertex disjoint cycles $C_{2k₁},...,C_{2kₗ}$, unless $G = K_{3,3} - 3K_{1,1}$