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        3
        votes
        0answers
        35 views

        Generalization of Menger's Theorem to Infinite Graphs

        Aharoni and Berger generalized Menger's Theorem to infinite graphs: For any digraph, and any subsets A and B, there is a family F of disjoint paths from A to B and a set separating B from A consisting ...
        0
        votes
        1answer
        57 views

        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 ...
        2
        votes
        0answers
        25 views

        Practical calculation of minimum weight vertex-disjoint cycle covers

        How are minimum-weight vertex-disjoint cycle covers of large dense symmetric graphs actually calculated in actual implementations? I know that the problem can be reduced to general matching by ...
        3
        votes
        0answers
        60 views

        Matching of two weighted graphs allowing one-to-many mapping

        I am looking for a heuristic for a graph matching problem as follows. Given two graphs: $A$ (consisting of nodes $a_i$) and $B$ (consisting of nodes $b_i$). Typically the size of $B$ is larger than ...
        -2
        votes
        2answers
        171 views

        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 $\...
        0
        votes
        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 ...
        3
        votes
        1answer
        75 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
        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 ...
        1
        vote
        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 ...
        2
        votes
        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 ...
        0
        votes
        1answer
        76 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
        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-...
        2
        votes
        0answers
        63 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
        143 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 ...

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