# Questions tagged [degree-sequence]

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17
questions

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25 views

### Degree-constraints for the existence of vertex-disjoint directed cycle covers in digraphs

Given a digraph $G(E,V): (u,v)\in E\implies(v,u)\notin E$, what is known about lower bounds on the indegree and outdegree of the vertices that guarantee the existence of a vertex-disjoint directed ...

**3**

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58 views

### Random graphs with prescibed degrees and triangles

In short: a random graph model generates (multi-)graphs with prescribed number of edges and minimal number of triangles for each vertex. Questions arise about the actual number of triangles and the ...

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80 views

### Minimum of sums over degree products in a directed acyclic graph

My problem is the following: we have a graph $ G=(V,E)$. Having a total ordering $ \eta $ of the nodes, we give a direction to the edges such that $ (u,v) $ is directed from $u$ to $v$ iif $ \eta(u) &...

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49 views

### Are two degree sequences compatible, for random simple graph generation?

Consider a set $V$ of $n$ vertices, and three degree sequences $a_i$, $b_i$ and $c_i$ such that $c_i = a_i+b_i$, $i=1..n$.
Assume these degree sequences are graphical: there exist simple graphs (no ...

**4**

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**3**answers

282 views

### Simple graphs with prescribed degrees as disjoint union of simple subgraphs with prescribed degrees

Consider a set $V$ of $n$ vertices, and three degree sequences $a_i$, $b_i$ and $c_i$ such that $c_i = a_i+b_i$, $i=1..n$.
Assume these degree sequences are graphical: there exist simple graphs (no ...

**3**

votes

**1**answer

85 views

### Degree sequences after vertex removals

Consider a graph $G=(V, E)$ with $|V|=n$ vertices. Let $(v_1, \dots, v_n)$ be an ordered list of its vertices. Let $G_i=G[\{v_{i+1}, \dots, v_n\}]$ be the induced subgraph on the last $n-i$ vertices. ...

**10**

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**4**answers

325 views

### When is a large graph with a given degree sequence likely to be connected?

Are there any results on whether a large random graph with a given degree distribution is likely to be connected?
In Erdős-Rényi graphs with $n$ vertices and $m$ edges, we have two sudden transitions ...

**7**

votes

**1**answer

325 views

### Construct a non-connected graph with a given degree sequence

Is there a known (efficient) algorithm to construct a non-connected graph with a given degree sequence (if it exists)?
Examples
The sequence $\{3, 2, 2, 2, 2, 2, 1\}$ has both connected and non-...

**6**

votes

**2**answers

136 views

### Eberhard-type theorems for Fisk triangulations?

A triangulation of a surface is called a Fisk triangulation if the degree of all but two vertices is even, and these two exceptional vertices of odd degree are neighbors.
I would like to know what ...

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108 views

### Irregularly Intertwined Linear Recursions: Other References?

I was wondering if anyone had run across the following notion of intertwined linear recursions. I'm looking for references, or even a standard name. (I know one source, which is the genesis of this ...

**5**

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**1**answer

162 views

### A simple requirement for a degree sequence to be graphical

The following theorem about the degree sequences of finite simple graphs is quite easy to prove from the Erdos-Gallai theorem.
Let $0 \lt \alpha \le \beta \lt n$ be integers. Call $(\alpha,\beta,n)...

**3**

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**0**answers

99 views

### Is there a Havel-Hakimi for geometric graphs?

Suppose that we are given $n$ points in the plane, with a degree prescribed for each, and the question is whether we can place a geometric graph on them. Is there an efficient algorithm for this?
...

**5**

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**2**answers

212 views

### Volume of the convex hull of the set of all graphic sequences of a given length

Consider the set of all graphic sequences with $n$ elements as a subset of $\mathbb{R}^{n}$, namely let
$$D(n)=\{(d_{1},\dots,d_{n})\in\mathbb{Z}_{+}^{n}:d_{1}\geq\dots\geq d_{n},\ \sum_{i=1}^{n}d_{i}\...

**2**

votes

**3**answers

257 views

### Making integer multisets graphic

Let $M=(X,f)$ be a multiset, where $X$ is the underlying set of elements and $f:X\rightarrow\mathbb{N}$ is the multiplicity function. For every $k\in\mathbb{N}$ put $k\cdot M:=(X,k\cdot f)$. It is ...

**8**

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**1**answer

947 views

### Smallest Connected Graph for Given Degree Sequence

For a given integer sequence $(d_1, d_2,...,d_n)$, a natural question is if such a sequence is graphical, i.e. is a degree sequence of some graph. According to Erdős–Gallai theorem, A sequence of non-...

**4**

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**1**answer

239 views

### Graphs with constant edge imbalance

The imbalance of an edge $(u,v) \in E(G)$ of a graph $G$ is defined as $|d(u)-d(v)|$ ($d$ being, as usual the degree). (This concept was introduced by Albertson in 1997)
I'm interested in the set of ...

**1**

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**1**answer

175 views

### Have you come across this kind of “degree” concept?

Let $A \subseteq V(G)$ be a set of vertices in a graph $G$ and let $v \in V(G)$ be some vertex. Define $d_{A}(v)$ as the number of neighbours of $v$ inside $A$.
Now suppose you have a graph whose ...