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Results tagged with graph-theory
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Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
7
votes
Accepted
Distinct closed walks with $2n$ steps in the $n$-dimensional hypercube
If you only wanted to quotient by symmetries of the cube, and not by shift (and presumably, reversal) of paths, there would be a very clean answer.
The symmetry group of the cube is $S_n \ltimes \{ \p …
- 156k
5
votes
Accepted
4-color theorem for hypergraphs
This follows from the case of graphs (i.e. hypergraphs where all sets have size $\leq 2$). As I explained in the comments above, Hadwiger's conjecture says that a graph with no $K_k$ minor is $(k-1)$- …
- 156k
9
votes
Accepted
Joining the $2^k$ points of $\{0,1\}^k$ with the shortest tree
I suspect that the optimum, for a cube of side length $2$, is $2^k \sqrt{3} - 2 \sqrt{3}+2$. Note that the optimum if we use edges of the cube is $2 (2^k-1)$, so this is better by a factor of roughly …
- 156k
6
votes
Accepted
When is the poset of acyclic orientations of a graph a lattice?
Vincent Pilaud's recent paper "Acyclic reorientation graphs and their lattice quotients" is a thorough answer to this question and every question like it. In particular, here is the answer to the part …
- 156k
5
votes
Accepted
Optimal schedule for a soccer tournament
Here is a streamlined description of the optimal strategies that have been found. In both strategies, there is a team that plays a special role -- call them $x$ -- and the other teams are numbered mod …
3
votes
Optimal schedule for a soccer tournament
Here is a first attempt, for others to improve on. For simplicity, I'll take $n=2k+1$ to be odd. We index the teams by integers modulo $n$.
We'll schedule $n$ rounds of $k$ games each. In the $j$-th r …
- 156k
4
votes
Accepted
Does the edge-graph of a centrally symmetric polytope determine which vertices are antipodal?
$\def\RR{\mathbb{R}}$
The answer is no! I will give an example of a centrally symmetric polytope in $\RR^4$ with $12$ vertices where there is a symmetry of the edge graph interchanging two non-antipod …
- 156k
1
vote
Number of tree walks of bounded degree
If we disregard $d$, we can get much more than $(\log k)^k$. Here is an easy construction to get $k^{k/2(1-\epsilon)}$, and a harder one for $k^{k(1-\epsilon)}$. I will also show that, if we hold $d$ …
- 156k
3
votes
When is the poset of acyclic orientations of a graph a lattice?
$\def\Acyc{\mathrm{Acyc}}$Here are some things I have figured out since asking the question. Thanks to John Machacek for pointing out that I should look at the literature on supersolvability and chord …
- 156k
3
votes
Accepted
Hamming representability of finite graphs
Yes. This is going to be very inefficient, but: Let $E$ be the number of edges and let $V$ be the number of vertices. I will embed $G$ into $H(|E|(|V|-1),\ 2|E|-2)$. To each vertex $v$ of $G$, we will …
- 156k
5
votes
Accepted
Can we realize a graph as the skeleton of a polytope that has the same symmetries?
A counterexample is giving by the edge graph of the dual to the polytope of Bokowski, Ewald and Kleinschmidt, described in Gil Kalai's answer. Let $P$ be the BEK polytope, let $P^{\vee}$ be its dual a …
- 156k
5
votes
Can the graph of a symmetric polytope have more symmetries than the polytope itself?
Partial progress: Let $V$ be the vertex set of $P$, let $E$ be the set of directed edges and let $X$ be the set of ordered pairs of distinct elements of $V$. Let $G$ be the group of combinatorial symm …
- 156k
5
votes
Non-isomorphic matroids with the same Tutte Polynomial
Matroids of ranks $1$ and $2$ have simple descriptions, from which one can check that the Tutte polynomial determines the matroid in these cases.
A matroid of rank $1$ is always $\ell \geq 0$ loops t …
- 156k
6
votes
Where does $2\sqrt{d-1}$ come from in Ramanujan graphs?
Heuristically, an expander $G$ looks locally like the $d$-regular tree $T$. Let $r$ be a positive real number and let $f_x$ be the function on the vertices of $T$ given by $f_x(y) = r^{d(x,y)}$. We ha …
- 156k
7
votes
Which paths in a graph are orthogonal to all cycles?
$\def\ZZ{\mathbb{Z}}$The answer to the boldfaced question is yes. Let $\Gamma$ be a connected graph with no cut edges. We will show that, if $\gamma \in C_1(\Gamma, \ZZ)$ with $\partial \gamma$ of the …
- 156k