# Questions tagged [bounded-degree]

The tag has no usage guidance.

20 questions
Filter by
Sorted by
Tagged with
1 vote
25 views

### Bounds on the problematic number of threads

Consider $n$ independent threads solving together a problem $P$. Thread-safety is here, roughly, a property of algorithms to solve the problem such that the concurrent execution of the problem does ...
• 111
123 views

### Hardnnes of Approximation of Minimum Vertex Cover on 3-Regular Graphs

The paper [Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs, Austrin, Khot, Safra] Shows that assuming the Unique Game Conjecture (UGC) the minimum vertex cover problem ...
• 412
71 views

### Is every 4-claw-free graph a bounded degree graph?

I am looking of some graph properties of 4-claw free graph, where neighborhood of every vertex has independent set of size at most 3. As per my observations, this type of independent set size ...
1 vote
54 views

### Maximum degree of the Sum of Squares certificate of a non-negative degree d polynomial on the boolean hypercube

Let $f: \{0, 1\}^n \rightarrow \mathbb R$ be a polynomial on the boolean hypercube. If $f$ is non-negative $(f \geq 0)$ i.e. $f(x) \geq 0, \forall x \in \{0, 1\}^n$ then $f$ always has a degree $2n$ ...
• 111
216 views

### NP-hardness: (planar) directed feedback vertex set problem with bounded degree

My question is the directed version of this one. (I know the results and proofs about feedback vertex set in undirected graphs or undirected planar graphs; so I am concern about the directed feedback ...
• 421
179 views

### Complexity of $(\Delta-1)$-coloring graphs of maximum degree $\Delta$

Suppose we have a graph $G$ with maximum degree $\Delta$. Deciding if $G$ can be colored with $\Delta$ colors is easy: Brooks' theorem says that this is always possible, unless $G$ is a clique or an ...
• 3,722
2k views

### Is the 3-coloring problem NP-hard on graphs of maximal degree 3?

Consider the 3-coloring problem: given an undirected graph $G = (V, E)$, decide if there is a 3-coloring of $G$, i.e., a function $f$ from $G$ to $\{1, 2, 3\}$ such that there is no edge $\{u, v\}$ in ...
• 9,697
1 vote
125 views

### Complexity of counting 3-colourings of planar bounded degree graphs

The following are known: It is #P-complete to count the number of 3-colourings of a planar graph (with respect to randomized reductions) [1]. For all $d\geq 3$, it is #P-complete to count the number ...
• 1,827
55 views

### The number of rooted ordered trees of max-out degree $k$

An ordered tree (also known as ordinal tree and plane tree) is a rooted tree in which the children of each node are ordered. It is known that the number of the ordered tree with $n$ edges is the $n$'...
152 views

### Sensitivity and Low-Degree Approximation under Non-Uniform Distribution

I am searching for generalizations of analysis of Boolean functions when the input strings are distributed according to a general non-uniform distribution, possibly with arbitrary dependencies between ...
1 vote
67 views

### relations between the degrees of a boolean function and its absolute function

Given a boolean function $f:\{0,1\}^n\rightarrow\mathbb{R}$ of degree $d$, is there any upper bound in terms of $d$ on the degree of the function $|f|$, where $|f|(x)=|f(x)|$. Here the degree of $f$ ...
378 views

### Finding subgraphs with high treewidth and constant degree

I am given a graph $G$ with treewidth $k$ and arbitrary degree, and I would like to find a subgraph $H$ of $G$ (not necessarily an induced subgraph) such that $H$ has constant degree and its treewidth ...
• 9,697
245 views

### Bias of a random boolean low degree polynomial

What is the bias of a random Boolean function that can be represented as a low degree polynomial over the reals, i.e. has low Fourier degree? More specifically, is it true that if we take a uniformly ...
• 1,927
470 views

### Degree restriction for polynomials in $\mathsf{VP}$

why do we need the restriction on the degree for a polynomial to be in VP due to which $(\prod_{i=0}^{n} x_i)^{2^n}$ $\notin$ VP even though it has a poly-size circuit?
• 327
510 views

### Random functions of low degree as a real polynomial

Is there a (reasonable) way to sample a uniformly random boolean function $f:\{0,1\}^n \to \{0,1\}$ whose degree as a real polynomial is at most $d$? EDIT: Nisan and Szegedy have shown that a ...
• 1,927
2k views

### hardness of approximating the chromatic number in graphs with bounded degree

I am looking for hardness results on vertex coloring of graphs with bounded degree. Given a graph $G(V,E)$, we know that for any $\epsilon>0$, it's hard to approximate $\chi(G)$ within a factor ...
• 271
188 views

### bounded outdegree bipartite spanners

Given an undirected graph $G = (V,E)$ and an integer $k > 0$, our objective is to find a subgraph $G' = (V ,E')$ where $E' \subseteq E$ such that $G'$ has the three following properties : $G'$ ...
• 644
2k views

### Is feedback vertex set problem solvable in polynomial time for 3-degree bounded graphs?

Feedback Vertex Set (FVS) is NP-complete for general graphs. It is known to be NP-complete for degree-$8$ bounded graphs due to a reduction from vertex cover. The Wikipedia article says that it is ...
• 368
270 views

### Complexity of finding if a degree bounded graph H is a subgraph of an unbounded graph G

You are given two graphs G and H , and want to know if H is a subgraph of G. You know that H has a max vertex degree K (constant integer). What can you say about the complexity of this? I know that ...
• 143