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        Questions tagged [perfect-matchings]

        A perfect matching is a matching of all the vertices of a graph. In other words, a perfect matching is a set of edges such that each vertex of the graph is incident to exactly one edge in the set.

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        Perfect matching for a particular type of bipartite graphs

        Let $G = (V_1 \cup V_2, E)$ be a connected and bipartite graph such that: (1) The number of vertices of $V_1$ is equal to the number of vertices of $V_2$, i.e., $|V_1| = |V_2| = N$ for some $N \in \...
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        Combining three matchings to form a maximal matching

        Consider a regular tripartite graph $G$ with maximum degree $\Delta\ge3$ and parts $A,B,C$. Now, the induced subgraphs $A\cup B, B\cup C$ and $A\cup C$ are all bipartite. Now, is there a way to ...
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        Cardinality of a set of mutually disjoint perfect matchings of $K_\omega$

        If $G=(V,E)$ is a simple, undirected graph, we say that $M\subseteq E$ is a perfect matching if the members of $M$ are pairwise disjoint and $\bigcup M = V$. Let $K_\omega$ be the complete graph on $\...
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        A vertex transitive graph has a near perfect/ matching missing an independent set of vertices

        Consider a power of cycle graph $C_n^k\,\,,\frac{n}{2}>k\ge2$, represented as a Cayley graph with generating set $\{1,2,\ldots, k,n-k,\ldots,n-1\}$ on the Group $\mathbb{Z}_n$. Supposing I remove ...
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        A simple case of a strong version of the Berge-Fulkerson conjecture

        UPDATE 28 June 2019 A counterexample for Conjecture 2 has been provided. The conjecture is now demoted again to guess. The text has been updated to reflect this change, and there is now a new ...
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        Perfect matchings and edge cuts in cubic graphs - part 1

        Let $G$ be a bridgeless cubic (simple) graph, and let $M$ be a perfect matching in $G$. $G-M$ will necessarily be a set of circuits. For example, if we delete a perfect matching from $K_{3,3}$ we ...
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        Biadjacency permanent upper bound in terms of genus of graph?

        Take $M$ to be biadjacency of a planar balanced bipartite on $2n$ vertices with genus $g$. Is it true for every $\epsilon\in(0,1)$ there is a $c_\epsilon>0$ such that $$\log\log(permanent(M))\leq\...
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        On statistics of perfect matchings between planar $4$ colorable and planar $3$ colorable

        Does the mean for number of perfect matchings of random graphs that are planar and $3$ colorable much higher than graphs that are planar are not $3$ colorable? Does a planar graph if $3$ colorable ...
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        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 ...
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        On optimal dual solutions for the minimum weight perfect matching problems in the case of metric weights

        Following Lovasz-Plummer (Matching theory, North-Holland 1986, Theorem 9.2.1), the minimum weight perfect matching problem on a complete graph $G$ with even number of vertices and weight $w:E(G)\to \...
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        Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs?

        Planar graph permanent can be reduced to determinants and so statistics should be amenable. Pick a uniformly random bipartite planar graph $G$ with $n$ vertices of each color and choose new ...
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        52 views

        Number of distinct perfect matchings/near perfect matchings in an induced subgraph

        Consider a Class 1 graph with degree $\Delta\ge3$ and the induced subgraph formed by deleting a set of independent vertices of cardinality $\left\lfloor\frac{n}{\Delta}\right\rfloor$. Then, what is ...
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        Do all induced subgraphs of powers of cycles have a perfect matching

        Do all independence induced subgraphs of powers of cycles have a distinct 1-factor? By independence induced, I mean those induced subgraphs which are formed by removing a maximal independent set of ...
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        Has this notion of vertex-coloring of graphs been studied?

        In a study of a quantum physics problem, I came about an apparently very natural type of vertex colorings of a graph. The colors of the vertex $v_i$ is inherited from perfect matchings $PM$ of an edge-...
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        237 views

        Minimum planar bipartite graph to cover all perfect matching count

        Given set $\mathcal T_n=\{0,1,\dots,2^n-1\}$ what is the minimum number of vertices $2m$ needed in a planar bipartite balanced graph such that at every $i\in\mathcal T_n$ there is a graph $G\in\...

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