# Tagged Questions

Vertex colouring, Edge Colouring, List Colouring, Fractional Chromatic Number and other variants of graph colouring problems are all on topic.

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### What is this subclass of $k$-colorable graphs called?

The following property emerged naturally when I was playing with certain generalizations of Kneser graphs. Let $k>0$ be a natural number. Consider a property $P_k$ of graphs defined as follows. ...
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### Can you prove that hypergraphs with n-1 edges are partially 2 colorable?

I can. But my proof uses a theorem (which I do not reveal yet to avoid influencing you) and it feels like an overkill, so I wonder if there is a simple proof. Now the problem. Suppose we have a ...
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### r-chromatic graphs with sufficient girth

For $k,l$ positive integers let $h(k,l)$ be the least integer with the property that in a graph on $h(k,l)$ vertices either there is a closed circuit of $k$ or fewer lines, or the graph contains $l$ ...
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### Representations of regular maps (four color theorem)

For the scope of the four color problem and without lack of generality, maps can be represented in different ways. This is generally done to have a different perspective on the problem. For example, ...
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### How many “different” colorings (excluding exchanges) exist for a given map (graph)?

In particular I'm interested in regular maps, excluding all maps that can be colored with 2 or 3 colors. For what I need to analyze, maps have to be regarded as differently colored, if the same ...
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### Why is “P vs. NP” necessarily relevant?

I want to start out by giving two examples: 1) Graham's problem is to decide whether a given edge-coloring (with two colors) of the complete graph on vertices $\lbrace-1,+1\rbrace^n$ contains a ...
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### code that produces all possible trees with n nodes. [closed]

I'm looking for code that produces all possible trees with no self edges (or their adjacent matrices) with n nodes, anyone have any idea if this is written anywhere?
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### Simplest examples of unique-solution and unsolvable-without-backtracking Sudoku-like problems

A The Sudoku game admits a broad generalization as follows : let $r$ be an integer $\geq 2$ and let $X$ be a finite set, and ${\cal X}$ be a collection of $r$-subsets of $X$ (i.e, a $r$-uniform ...
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### Is the chromatic number of the real plane invariant under the norm?

Recall that chromatic number of $\mathbb{R}^2$ is the least $n$ such that there exists a function $f$ from $\mathbb{R}^2$ into a set of colors ${C_1,\ldots,C_n}$ with $f(x)\neq f(y)$ for $||x-y||_2=1$....
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### Edge-coloring of a graph - special case

Hi everyone, I have a problem I am working on that can be reduced to the following special case of edge coloring. Let $G = (V,E)$ be an arbitrary graph. Furthermore, let each edge be assigned a ...
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### 4-coloring maps of pentagons

Is there a simple proof for the 4-color theorem when restricted to (finite) maps all of whose regions are pentagons? I am in fact most interested in convex pentagons, if that additional structure ...
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### Coloring edges on a graph s.t. the set of edges for any two vertices have no more than 'k' colors in common

Please imagine the case where one has a planar graph, $G$, with a set of $|V|$ vertices, $(v_1, ..., v_{|V|}) \in V$, and $|E|$ edges, $(e_1, ..., e_{|E|}) \in E$. Now, provided a total of $N$ colors,...
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### Labelled spanning trees without edge crossings

Draw the complete graph $K_n$ on a plane in general position with every edge a straight line and randomly label the edges $0$ or $1$. Does this graph always have a spanning tree with no edges crossing ...
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### Hypergraph Chromatic Number vs Degree, Clique-Size

For a hypergraph let $\chi$ be the least number of colours needed to colour the vertices, so that in each edge, each colour is used at most once (i.e., the strong chromatic number). Let $\Delta$ be ...
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### finding set of tree decompositions to cover all pairs of vertices

I first asked this on cstheory.SE but got no reply. Let $P(X_i=x)$ represent probability that randomly chosen proper $q$-coloring of an $L\times L$ square grid contains color $x$ at position $i$. How ...
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### Derivative of Tutte polynomial at -1

Let Tutte polynomial on graph with edge-set $E$ be defined as follows $$f(q,v)=\sum_{A\subseteq E} q^{\kappa(A)} v^{|A|}$$ Here the sum is over all subgraphs $A$, $\kappa(A)$ is the number of ...