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3-biplacement of bipartite graphs

Adamus L., Leśniak E., Orchel B.

Opuscula Mathematica

2008

Vol. 28, no. 3

223-231

Let G = (L,R;E) be a bipartite graph with color classes L and R and edge set E. A set of two bijections {φ1, φ2}, φ1, φ2 : L ∪ R → L ∪ R, is said to be a 3-biplacement of G if [formula], where φ*/1, φ*/2 are the maps defined on E, induced by φ1, φ2, respectively. We prove that if ‌L‌ = p, ‌R‌ = q, 3 ≤ p ≤ q, then every graph G = (L, R; E) of size at most p has a 3-biplacement.

Bipartite embedding of (p, q)-trees

Orchel B.

Opuscula Mathematica

2006

Vol. 26, no. 1

119-125

A bipartite graph G = (L, R; E) where V(G) = L ∪ 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 ⊆ L', R ⊆ R' and E ⊆ E'. In this paper we present sufficient degree conditions for a bipartite graph to contain a (p, q)-tree.

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

Orchel B.

Opuscula Mathematica

2005

Vol. 25, no. 2

269-274

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.