# Questions tagged [co.combinatorics]

Questions related to combinatorics and discrete mathematical structures

169 questions with no upvoted or accepted answers
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### How good is greedy in average?

Given a family ${\cal F}\subset 2^E$ of (feasible solutions), the maximization problem on ${\cal F}$ is, for every weighting $x:E\to \{0,1,\ldots\}$ of ground elements, to compute the maximum weight ...
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### Is RAMSEY COLORING in $NC$?

Say that a function $f$ is a RAMSEY COLORING if on unary input $n$, it returns a complete graph on $n$ vertices with its edges colored red and blue without a monochromatic clique of size $10\log n$. ...
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### Random unbalanced bipartite graphs are good small set expanders

My question is about small set expansion properties of random unbalanced bipartite graphs. Fix a positive $\delta<1/2$, and a positive integers $n,m,d$. Let us call a bipartite graph $\mathcal{G}$...
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### Complexity of checking the equivalence of constraints and generator descriptions of convex polyhedra

Consider the space $\mathbb{Q}^n$. A convex polyhedron is defined, equivalently, by a system of linear (in)equalities (with integer coefficients) or by a system of generators: vertices, and in case ...
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### How to prove "obvious" facts?

The title is somewhat "arrogant": say, most of us treat $P\neq NP$ as an "obvious" fact, albeit no proof is in sight. But my question is at a much, much lower level, is about a fact which "should be" ...
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### Computing weighted sums of binomial coefficients

This question is a reformulation of Complexity for computing weighted number of paths on integer lattice Is there any way to compute in $o(n^2)$ all $n$ sums $\sum_{0\leq i \leq j} a_i\binom{j}{i}$ ...
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### Statistical relationship between diameter and density in strongly connected random digraphs

I would like to know if there is any known relation between diameter and density in (connected) simple digraphs. Asymptotic results for n→∞, statistical, conditioned results that would restrict the ...
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### Spectral gap for random bipartite regular graphs

For a graph $G$, let its Laplacian be $\Delta =I − D^{−1/2}AD^{−1/2}$, where $A$ is the adjacency matrix, $I$ is the identity matrix and $D$ is the diagonal matrix with vertex degrees. I'm ...
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### Upper Bound on Number of $n \times n$ Boolean matrices of Boolean rank at most $k$

An $n \times n$ Boolean matrix $B$ has Boolean rank $k$ if there exist matrices $L \in \{0,1\}^{n \times k}$ and $R \in \{0,1\}^{k \times n}$, s.t. $B = L \circ R$. Here $\circ$ denotes the Boolean ...
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### Approximating a max-cut's intersection with other cuts

For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set $S$ and its complement. A max cut is one with at least as many edges as any ...
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### Counting small terms in a determinant calculation over polynomials (counting spanning trees by weight)

I have a $n\times n$ matrix $A$. It's terms are $a_{ij}=-x^{w_{ij}}$ if $i\neq j$ and $a_{ii}=\sum_{j=0}^{n+1} x^{w_{ij}}$ on the diagonal. The matrix is symmetric as $w_{ij}=w_{ji}$. Numbers $w_{ij}$ ...
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### Extension of Cheeger's inequality with distinguished vertices

The standard Cheeger's inequality for graph $G$ states that $\frac{1}{2}$ $\lambda$ < $\phi(G)$ < $\sqrt{2\lambda}$ where $\lambda$ is the second smallest eigenvalue of the normalized ...
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### Existence of "colouring matrices" — a generalisation

This is a generalisation of the following post: Existence of "colouring matrices". As the base case turned out to be fairly straightforward (in essence, precisely equal to the existence of Sperner ...
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### Lossless, constant-degree expanders that expand large sets

It is known how to construct "lossless" unbalanced bipartite expanders with the following properties: the bipartite graph has $n$ left vertices, $m$ right vertices, left-degree $D$, and for all left-...
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### Is this graph polynomial known? Can it be efficiently computed?

Consider a connected simple graph $G$ with $n$ vertices and $m$ edges. View each edge $\ell$ as a transposition $t_{\ell}$ acting on the set of vertices. [To be more explicit, given an edge $\ell$ ...
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### General Results for Complicated Constraint Satisfaction Problem

Consider the following problem: on a finite two-dimensional grid (say the grid points are vertices of a graph), I need to color the vertices in such a way to satisfy the existence and nonexistence of ...