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Treewidth related properties of a bipartite graph with bounded local crossing number and diameter

If a bipartite degree at most $3$ graph on $O(n^2)$ vertices with diameter at most $O(\log n)$ has property that every edge intersects at most $O(\log n)$ edges on a planar drawing then does any of ...
47 views

Partitioning vertex set to maximize weights of inter-class edges?

An interesting problem has come up in my work, and I haven't quite been able to find references to it so I thought I'd ask here. Suppose we have some complete, weighted graph with vertex set $V$. Is ...
53 views

Maximum number of perfect matchings in a graph of genus $g$ balanced $k$-partite graph

What is the maximum number of perfect matchings a genus $g$ balanced $k$-partite graph (number of vertices for each color in all possible $k$-colorings is within a difference of $1$) can have? I am ...
164 views

For any topological space $(X,\tau)$ we define a matching to be a collection of non-empty and pairwise disjoint open sets. We define the matching number $\nu(X,\tau)$ to be the smallest cardinal $\... 2answers 70 views On a condition concerning the number of neighbors in bipartite graphs For any undirected simple graph$G=(V,E)$we define for$v\in V$the set$N(v) = \{w\in V: \{v,w\}\in E\}$. Suppose$A, B$are finite, disjoint sets, and$G = (A\cup B, E)$is a bipartite graph with ... 0answers 116 views Is the partition of bipartite graphs NP-hard? I wonder if the following problem is NP-hard. Is it? Given a bipartite graph$G = (U, V, E)$with weights$w : E \to \mathbb{R}_+$, find a partition of$U$into$U_1, U_2$and nonempty disjoint ... 1answer 87 views Maximum partition of bipartite graph Let$G = (U, V, E)$be a bipartite graph. Let$w: E \to \mathbb{R}$be a weight function on the edge set$E$. Given subsets$U_1,\ldots, U_k \subset U, U_i\cap U_j = \emptyset$and a partition$V_1,\...
249 views

Algorithm to find a $k$-partite graph

Is there any algorithm which finds any $k$-partite graph of a given graph which is known to be a $k$-partite graph? For example, you are given a graph $G$ with vertices $V$ and edges $E$, and you ...
443 views

Combinatorial optimization problem for bipartite graphs

Let $G(V_1\cup V_2, E)$ be a simple bipartite graph having $n$ vertices and $m$ edges, such that $|V_1|=|V_2|$ (which implies that $n$ is an even number). Given any node $i \in V_1\cup V_2$, we denote ...
171 views

How many $40$-vertex cubic bipartite graphs have determinant $\pm 3$?

To get some feel for the size of a particular computation, I would like to know the approximate number of (pairwise-nonisomorphic) cubic bipartite graphs on $40$ vertices whose bipartite adjacency ...
104 views

Existence of bipartite subgraphs satisfying degree and edge cardinality constraints

How can we prove the following conjecture? Given any simple unweighted bipartite graph $G(V_1, V_2, E)$, there always exists a subgraph $G'(V_1, V_2, E')$ of $G$ such that the two following ...
167 views

Given an integer $k>1$, is there a connected bipartite graph $\Gamma = (A, B, E)$ where $A\cap B = \emptyset$ and $E \subseteq \big\{\{a, b\}:a\in A, b\in B\big\}$ such that $|A| = |B|$, $\text{... 3answers 297 views Hamiltonian paths in bipartite graphs with 2 sets of “almost” same cardinality Suppose we have two finite disjoint sets$A, B \neq \emptyset$such that$|A|$and$|B|$differ by at most$1$, and let$\Gamma = (A\cup B, E)$where$E\subseteq \big\{\{a,b\}: a\in A, b\in B\big\}$... 1answer 370 views Graph to Bipartite conversion preserving number of perfect matchings Given a graph$G$on$n$vertices is there a technique to convert to a balanced bipartite graph$B$with$O(n^c)$vertices at some fixed$0<c$in$O(n^{c'})$time at some fixed$0<c'$such that ... 0answers 142 views Minimal size of the maximum biclique We examine a bipartite graph with two sides$R$and$L$, and denote by$|L|$and$|R|$the number of nodes in each side. We know only that each vertex on side$R$is connected to$k$vertices on side$...

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