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# All Questions

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### Algorithmic complexity of deciding the existence of regular $\mathrm{f}$-factors in graphs

Finding regular $\mathrm{f}$-factors in undirected simple graphs can be reduced to finding a perfect matching by utilizing the gadgets of Tutte or of Lovász and Plummer; there are several algorithms ...
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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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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 $\... 1answer 65 views ### A vertex transitive graph has a near perfect/ matching missing an independent set of vertices Consider a power of cycle graph$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$. Supposing I remove ... 0answers 90 views ### A simple case of a strong version of the Berge-Fulkerson conjecture UPDATE 28 June 2019 A counterexample for Conjecture 2 has been provided. The conjecture is now demoted again to guess. The text has been updated to reflect this change, and there is now a new ... 0answers 22 views ### Perfect matchings and edge cuts in cubic graphs - part 1 Let$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}$we ... 1answer 53 views ### Maximum number of perfect matchings in a graph of genus$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 ... 1answer 55 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 319 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 973 views ### Vertex coloring inherited from perfect matchings (motivated by quantum physics) Added (24.08.2019): As I consider this question important for quantum physics, I have announced a 3000 Euro award on its solution, see here for more details. Added (19.09.2019): This problem can be ... 1answer 187 views ### Why is the number of Perfect Matchings in a triangular grid equivalent to the number of Royal Paths? The sequence A006318 at OEIS stands for the Schr?der numbers. They describes the number of lattice paths from the southwest corner$(0,0)$of an$n\times n$grid to the northeast corner$(n,n)$, ... 0answers 33 views ### Can Orientability of Manifolds be Generalized to TSP Instances? It is well known, that there are two basic kinds of manifolds, orientable and non-orientable ones; the most simple examples being obtained by identifying a pair of opposite sides of a rectangular ... 1answer 109 views ### Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs? In OEIS A003436, it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional Octahedron is the same as the number of Perfect Matchings in a the complement of the Cycle ... 1answer 466 views ### Graphs with only disjoint perfect matchings, with coloring The following purely graph-theoretic question is motivated by quantum mechanics. Definitions: A bi-colored graph$G\$ is an undirected graph where every edge is colored. An edge can either be ...

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