# All Questions

Tagged with graph-theory matching-theory

69
questions

**0**

votes

**1**answer

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

**1**answer

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**

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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**

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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**

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**1**answer

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**

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**1**answer

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**

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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-...

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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**

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**1**answer

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?

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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**

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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

**2**answers

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

**2**answers

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 $C\cap e \neq \emptyset$...

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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**

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**1**answer

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 ...