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      59 views

      Proving Vizing's and Brooks' theorem using the polynomial approach

      It is known that the graph polynomial defined by $\prod_{i<j}(x_i-x_j)$ where the vertices $x_k\ \ , \ \ k=\{1,2\ldots,n\}$ are ordered with respect to some order; can be used to verify the proper ...
      4
      votes
      1answer
      116 views

      Graphs with Hermitian Unitary Edge Weights

      Very recently, Hao Huang proved the Sensitivity Conjecture, which had been open for 30 years or so. Huang's proof is surprisingly short and easy. Here is Huang's preprint, a discussion on Scott ...
      3
      votes
      2answers
      96 views

      Strong chromatic index of some cubic graphs

      Edit 2019 June 26 New computer evidence forces us to revise our guesses relating strong chromatic index and girth Edit 2019 June 25 Some mistakes have been corrected. Question 2 has changed. ...
      2
      votes
      0answers
      17 views

      Complexity of weighted fractional edge coloring

      Given an edge-weighted multigraph $G=(V,E)$ with a positive, rational weight function $(w(e): e \in E)$, the weighted fractional edge coloring problem (WFECP) is to compute ($\min 1^T x$ subject to $...
      1
      vote
      0answers
      51 views

      Succinct circuits and NEXPTIME-complete problems

      I am fascinated by a recent fact I was reading: Succinct Circuits are simple machines used to descibe graphs in exponentially less space, which leads to the downside that solving a problem on that ...
      3
      votes
      1answer
      124 views

      Representation of Subgraph Counts using Polynomial of Adjacency Matrix

      We consider a graph $G$ of size $d$ with adjacency matrix $A$, whose entries take value in $\{0,1\}$. We are interested in the number of a certain connected subgraph $S$ of size $k$ in $G$. For ...
      -4
      votes
      1answer
      197 views

      What is the computationally simplest way to universally index the set of simple graphs?

      If given a simple, integer-labeled, but not necessarily connected, graph $G := (V,E)$ consisting of at least one vertex, i.e. $\lvert \rvert V \lvert \rvert \geq 1$, then is there a function to ...
      1
      vote
      1answer
      58 views

      The complexity on calculation of the Graev metric on the free Boolean group of a metric space

      For a set $X$ by $B(X)$ we denote the family of all finite subsets of $X$ endowed with the operation $\oplus$ of symmetric difference. This operation turns $B(X)$ into a Boolean group, which can be ...
      2
      votes
      0answers
      31 views

      Flat or linkless embeddings of graph with fixed projection

      The problem of finding whether a given planar diagram of a graph, with over- and under-crossings, is a linkless embedding or not has unknown complexity (Kawarabayashi et al., 2010). My first question ...
      0
      votes
      1answer
      83 views

      Is there any solution that currently exists for the graph automorphism problem in the general case?

      I was reading the Wikipedia pages on the graph automorphism, but I could not find any solution to the problem (Not even a brute force one). So, is it indeed true that no solutions exist for the ...
      1
      vote
      0answers
      31 views

      Is there an algorithm for this constrained Hypergraph optimization problem?

      I'm currently developing an algorithm for computing knot coloring invariants and got to the following question: Given a set $S$ and a certain hyper-graph $H \subseteq S^3 $, find a decomposition $S = ...
      3
      votes
      0answers
      45 views

      Karp hardness of two cycles which lengths differ by one

      Our problem is as follows: NEARLY-EQUAL-CYCLE-PAIR Input: An undirected graph $G(V,E)$ Output: YES if there exists $2$ (simple) cycles in $G$ which lengths differ by $1$, otherwise NO Is ...
      6
      votes
      1answer
      138 views

      What is the complexity of counting Hamiltonian cycles of a graph?

      Since deciding whether a graph contains a Hamiltonian cycle is $NP$-complete, the counting problem which counts the number of such cycles of a graph is $NP$-hard. Is it also $PP$-hard in the sense ...
      7
      votes
      0answers
      109 views

      Does the problem of recognizing 3DORG-graphs have polynomial complexity?

      A 2DORG is the intersection graph of a finite family of rays directed $\to$ or $\uparrow$ in the plane. Such graphs can be recognized effectively (Felsner et al.). A 3DORG is the intersection graph of ...
      6
      votes
      0answers
      85 views

      Combinatorial region-halfplane incidence structures

      I've seen a bunch of similar MO questions, yet hopefully this is not a complete duplicate. Consider $n$ halfplanes in $\mathbb{R}^2$ with their borders in general position, that is, no point of $\...

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