‎for a set of non-negative integers~$l$‎, ‎the $l$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

‎For a set of non-negative integers~$L$‎, ‎the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$‎. ‎The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...

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.

We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers k, l the notions of (k, l)-rigid and (k, l)-stress free bipartite graphs. This theory coincides with the study of Babson–Novik’s balanced shifting restricted to graphs. We establish bipartite analogs of the cone, contraction, deletion, an...

A m,n -bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell D n, a, b consists of the path Pn−a−b together with a independent vertices adjacent to one pendent vertex of Pn−a−b and b independent vertices adjacent to the other pendent vertex of Pn−a−b. In this paper, firstly, we show that, among m,n bipartite gra...

In this note, we give tighter bounds on the complexity of the bipartite unique perfect matching problem, bipartite-UPM. We show that the problem is in C=L and in NL , both subclasses of NC. We also consider the (unary) weighted version of the problem. We show that testing uniqueness of the minimum-weight perfect matching problem for bipartite graphs is in L= and in NL. Furthermore, we show that...

Online Bipartite Matching is a generalization of a well-known Bipartite Matching problem. In a Bipartite Matching, we a given a bipartite graph G = (L,R,E), and we need to find a matching M ⊆ E such that no edges in M have common endpoints. In the online version L is known, but vertices in R are arriving one at a time. When vertex j ∈ R arrives (with all its edges), we need to make an irreversi...

Fouquet, J.-L. and A.P. Wojda, Mutual placement of bipartite graphs, Discrete Mathematics 121 (1993) 85-92. Let G =(L, R, E) and H =(L’, R’, E’) be bipartite graphs. A bijection 4: L w R + L’v R’ is said to be a biplacement of G and H if 4(L)= L’ and ~$(x)~$(y)gE’ f or every edge xy of G. A biplacement of G and its copy is called a 2-placement of G. We prove that, with some exceptions, every bi...

The bipartite vertex (resp. edge) frustration of a graph G, denoted by ψ(G) (resp. φ(G)), is the smallest number of vertices (resp. edges) that have to be deleted from G to obtain a bipartite subgraph of G. A sharp lower bound of the bipartite vertex frustration of the line graph L(G) of every graph G is given. In addition, the exact value of ψ(L(G)) is calculated when G is a forest.

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...