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        1answer
        58 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 ...
        3
        votes
        1answer
        67 views

        Equitable edge coloring of graphs

        Consider a simple regular graph $G$ with $n$ vertices and $E$ edges. Then, can we say that the edges can be colored equitably in $\Delta+1$ colors? By equitability is meant that a proper $\Delta+1$ ...
        1
        vote
        0answers
        78 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 ...
        1
        vote
        0answers
        19 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 ...
        2
        votes
        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 ...
        0
        votes
        1answer
        65 views

        All even order graphs with $\Delta\ge\frac{n}{2}$ is Class 1

        Are all even order graphs with maximum degree $\ge\frac{|V(G)|}{2}$ Class 1(edge-colorable(chromatic index) with $\delta(G)$ colors)? Here, $|V(G)|$ detnotes the number of vertices in the graph. I ...
        6
        votes
        0answers
        267 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-...
        2
        votes
        0answers
        61 views

        Graph pattern matching

        Given a weighted, oriented, connected graph with $10^7$ vertices and $10^{10}$ edges I need to implement the algorithm for searching various patterns on this graph for less than polynomial time. ...
        2
        votes
        1answer
        141 views

        Graphs with exactly $n$ perfect matchings

        Is there for every $n\in\mathbb{N}$ a connected, simple, undirected graph $G_n=(V_n,E_n)$ such that $G_n$ has exactly $n$ perfect matchings?
        0
        votes
        1answer
        85 views

        Connected infinite graphs in which all matchings are “small”

        Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all matchings $M\subseteq E$ the set $V\setminus (\bigcup M)$ is ...
        3
        votes
        1answer
        80 views

        Maximal matchings in connected graphs

        Let $n\in\mathbb{N}$ be a positive integer. Is there a connected graph $G$ such that $G$ cannot be coloured with less than $n$ colours, and every two maximal matchings have non-empty intersection? (I ...
        2
        votes
        2answers
        106 views

        Graphs in which all maximal matchings intersect

        Let $G=(V,E)$ be a simple, undirected graph. A matching is a set $M\subseteq E$ consisting of pairwise disjoint edges. We say $M$ is maximal if it is maximal amongst all matchings in $G$ with respect ...
        4
        votes
        2answers
        82 views

        Vertex cover number vs matching number

        Let $G=(V,E)$ be a finite, simple, undirected graph. A matching is a set $M\subseteq E$ of pairwise disjoint edges. A vertex cover is a set $C\subseteq V$ of vertices such that $C\cap e \neq \emptyset$...
        1
        vote
        0answers
        61 views

        Matchings in infinite, not necessarily bipartite, graphs

        Aharoni, Nash-Williams, and Shelah have extended the famous marriage theorem for finite bipartite graphs due to Hall to arbitrary graphs. Is there a similar generalization of Tutte's theorem on ...
        0
        votes
        1answer
        107 views

        A weaker version of Dirac's theorem

        This is related to Dirac's theorem. For any finite, simple, undirected graph $G=(V,E)$ let $\delta(G)$ denote the minimal degree of all vertices. Are there positive integers $n,c\in\mathbb{N}$ with ...

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        山西福彩快乐十分钟
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