# Questions tagged [bounded-degree]

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### 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 $2n$ ...
194 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 ...
161 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 ...
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 ...
1 vote
117 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) . For all $d\geq 3$, it is #P-complete to count the number ...
54 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 $n$'...
150 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
61 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 $f$ ...
365 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 ...
229 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 ...
444 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?
466 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 ...
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 ...
187 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 : $G'$ ...
1k 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 ...