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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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### 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 ...
34 views

### 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 ...
52 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 ...
137 views

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 $w:E(G)\to \... 0answers 30 views ### Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs? Planar graph permanent can be reduced to determinants and so statistics should be amenable. Pick a uniformly random bipartite planar graph$G$with$n$vertices of each color and choose new ... 1answer 52 views ### Number of distinct perfect matchings/near perfect matchings in an induced subgraph Consider a Class 1 graph with degree$\Delta\ge3$and the induced subgraph formed by deleting a set of independent vertices of cardinality$\left\lfloor\frac{n}{\Delta}\right\rfloor$. Then, what is ... 0answers 35 views ### Do all induced subgraphs of powers of cycles have a perfect matching Do all independence induced subgraphs of powers of cycles have a distinct 1-factor? By independence induced, I mean those induced subgraphs which are formed by removing a maximal independent set of ... 0answers 315 views ### Has this notion of vertex-coloring of graphs been studied? In a study of a quantum physics problem, I came about an apparently very natural type of vertex colorings of a graph. The colors of the vertex$v_i$is inherited from perfect matchings$PM$of an edge-... 1answer 237 views ### Minimum planar bipartite graph to cover all perfect matching count Given set$\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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