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        All Questions

        3
        votes
        1answer
        42 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 ...
        1
        vote
        1answer
        48 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 ...
        5
        votes
        1answer
        161 views

        “König's theorem” for $T_2$-spaces?

        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 $\...
        -1
        votes
        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 ...
        1
        vote
        0answers
        113 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 ...
        0
        votes
        1answer
        81 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,\...
        1
        vote
        1answer
        176 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 ...
        4
        votes
        1answer
        425 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 ...
        5
        votes
        3answers
        163 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 ...
        2
        votes
        1answer
        98 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 ...
        1
        vote
        2answers
        157 views

        Hamiltonicity and minimal degree in bipartite graphs

        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{...
        0
        votes
        3answers
        258 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 $E\subseteq \big\{\{a,b\}: a\in A, b\in B\big\}$ ...
        7
        votes
        1answer
        341 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 ...
        2
        votes
        0answers
        139 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 $...
        2
        votes
        1answer
        135 views

        Minimal size of the maximal 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 node on side $R$ is connected to $k$ nodes on side $L$, ...

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