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10

It depends entirely on what level of formality you're aiming for. The informal description of an algorithm in your question is quite enough to convince me that 3-colourability is decidable. If you wanted to be a bit more formal, you could give pseudocode. If you wanted to be more formal still, you could describe a Turing machine in English. If you wanted to ...

8

You must remember that the vertices diagonal from one another can be colored the same! Your formula does not take that into account. We can find the chromatic number of a graph via the inclusion-exclusion principle. It is a very general counting technique that allows us to count complex structures, if we can prove certain bounds on certain subsets. The ...

7

As far as we know, there is no simple characterization for 3-colorability. Deciding if a given graph is 3-colorable is $\sf NP$-complete. However, we know plenty of structured graph classes for which the problem is easy. For example, Grötzsch's theorem states every triangle-free planar graph is 3-colorable. Furthermore, such graphs can be 3-colored in linear ...

7

One of the 4 Color Theorem most notable applications is in mobile phone masts. These masts all cover certain areas with some overlap meaning that they can’t all transmit on the same frequency. A simple method of ensuring that no two masts that overlap have the same frequency is to give them all a different frequency. But, as the government owns all ...

7

The labels $c_v$ and $c_p$ are relative. So when a node (parent in your example) having $c_v = 1010010000$ receives from its parent (grandparent in your example) an id $c_p = 0010110000$, the difference, as you correctly point out, is in the fifth position. Now, the total number of bits in the original ids is 10, so representing any index (0-9) will require ...

7

Whenever you are interested in a (well-known) graph invariant, it's a good idea to check out ISGCI first. Have a look at graphs that have bounded chromatic number, or graphs for which computing the number is doable in polynomial time. To shortly summarize the above, one could highlight perfect graphs (a superclass of chordal graphs), and graphs of bounded ...

7

Let us assume that the dual graph is connected, which means that if you connect any two faces which share an edge, then you get a connected graph on the triangular faces. Pick an arbitrary triangular face $F$ and color its 3 vertices with 3 distinct colors. If $F'$ shares an edge with $F$, then there is only one choice for the color of the remaining vertex ...

6

Your problem can be solved in linear time, by reduction to 2SAT. For each vertex $v$ we will have three variables $v_R,v_B,v_G$ and clauses $\lnot v_R \lor \lnot v_B,\lnot v_R \lor \lnot v_G, \lnot v_B \lor \lnot v_G$. These ensure that at most one of $v_R,v_B,v_G$ is true. For each edge $(v,w)$ labeled $R$, we will add the clause $v_R \lor w_R$. If there is ...

6

The idea is to use dynamic programming. For every pair of vertices $x,y$ and subset $S$ of colors, you determine whether there is a path from $x$ to $y$ of length $|S|$ (measured in vertices) which uses all colors in $S$, each of them exactly once. What is this good for? Usually our graphs are not colored. The idea of color coding is that finding cycles (...

5

No, you couldn't say that, because it's not true. A helpful method is to prove all your claims. Don't just guess -- try to find a proof. If you're struggling to find a proof, the first thing to check is whether maybe the thing you're trying to prove is false, so look for a counterexample. In this case you should easily be able to find a counterexample to ...

5

You're right that the statement is false. The correct statement states that every undirected simple graph in which each node has at most $d$ neighbors can be colored using $d+1$ colors so that each edge connects nodes having different color. Stated in more idiomatic language, every undirected simple graph of maximum degree $d$ can be (properly) colored using ...

5

Many existing heuristics for graph coloring can work even if you specify the colors of a few vertices. So, here is one plausible algorithm you could use: We are given an existing coloring $C$. Pick two vertices $v,w$ randomly. We are going to assign colors for $v,w$ (in the new coloring), leave the other vertices unassigned, and use some existing graph ...

5

For an instance of 3-COLOR, try to add a complete graph of size $k-3$, and add an edge between each new vertex and each old vertex. Now you can prove the new graph is $k$-colorable iff the old graph is 3-colorable.

5

So how would one generate RGB vectors with a constant norm that are still valid RGB values? ($0 \le r,g,b \le 255$) There is a simpler algorithm. Let $L$ be the given norm. Verify that $0\le L\le 255\sqrt3$. Let r be a random number between $\sqrt{L^2-\min(2\cdot255^2, L^2)}$ and $\min(255, L)$. Let $M=\sqrt{L^2-r^2}$. Let g be a random number between $\... 4 Hint: a graph is not bipartite if there is a walk of odd length from a vertex to itself. 4 Since graph 2-coloring is in P and it is not the trivial language ($\emptyset$or$\Sigma^*$), it is NP-complete if and only if P=NP. 4 You can of course take any graph class for which coloring is easy, and additionally require that the maximum degree is at most 4. For example, every bipartite graph of maximum degree at most 4 works. Or "bipartite" could be replaced with say "outerplanar" or more generally "small cliquewidth". But having maximum degree at most 4 alone will not work. That is,... 4 Generate a random three-dimensional vector$v$with non-negative components (you can generate each of the three components at random). Then, fix up the norm to be$L$by setting $$w = L \cdot {v \over \|v\|}.$$ Now$w$will have norm$L$. Interpret$w$as a RGB color. Next, check that all three components of$w$are in the range 0..255. If yes, you've ... 4 One can view this problem as a dynamic programming problem with$3N$subproblems. Let$RR(N)$be the number of solutions for a$2\times N$matrix where the first row is colored with red-red,$RB(N)$the number of solutions where the top cell is red and the bottom one blue, and$BR(N)$the number of solutions where the top cell is blue and the bottom one red.... 4 I will show you how you can improve the computational complexity of Tom's solution. Let's rewrite his recursive relationship: $$RR(N) = RR(N - 1) + 2BR(N - 1)$$ $$BR(N) = RR(N - 1) + BR(N - 1)$$ You can express this relationship using matrix multiplication.$ \left( \begin{array}{cc} RR(N) \\ BR(N) \end{array} \right) % = \left( \begin{array}{cc} 1 & ...

3

A major part of proving that a problem is NP-hard by reduction is proving that the reduction works. It's not enough to find some procedure which takes an instance of problem A and transforms it to an instance of problem B; you have to figure out the connection betweens solutions of the one instances and solutions of the other. In your case, one idea that ...

3

A problem $P$ being NP-hard means that all problems in NP can be reduced to $P$. That's by definition. Working out the exact reduction needed can be tricky, which is why one usually looks for a somehow similar problem to prove hardness. You probably wouldn't try to directly reduce MINIMUM INTERVAL GRAPH COMPLETION to Tetris to prove its hardness. In ...

3

It is $NP$-complete. Consider a graph $G$ which is modified by duplicating every vertex, and connecting every duplicate vertex to its original. Then if we constrain all the duplicate vertices to a fixed color, then the thus obtained graph is 4-colorable (with constraints) if and only if the original graph is 3-colorable.

3

Graph coloring problems are widely applicable to the problem of scheduling. Consider a University, where you are trying to schedule times for all of the final exams. Some students are taking more than one class, so you want to make sure they don't have two exams scheduled at the same time. However, you want your exam writing period to be as short as ...

3

Hajós came up with a calculus for proving that graphs are not 3-colorable. The calculus is complete, in the sense that every non-3-colorable graph can be proved to be non-3-colorable. Pitassi and Urquhart related the strength of this proof system to the classic Extended Frege proof system: if one system is optimal (has short proofs for all true statements), ...

3

For each value $k\in \mathbb{N}$, any graph that has a $(k+1)$-clique is not $k$-colorable. So a 4-clique rules out the existence of a 3-coloring. Be aware, that this is only a necessary condition (i.e. there are graphs without a 4-clique that are still not 3-COL), while the odd-length cycle condition for 2-colorings is also sufficient (i.e. any graph ...

3

If $|M| = 2$, this is the max-cut problem.

3

[...] is this feasible in a real world scenario? I'm not sure. But note that determining the chromatic number of a graph and determining the minimum number of timeslots needed to schedule lectures with a given pattern of conflicts are exactly the same problem. So, if the scheduling problem can't be feasibly solved by colouring, it can't be feasibly solved ...

3

You can prove an even stronger claim. Consider the following algorithm: Let $v_1,\ldots,v_n$ be an ordering of the vertices such that for $i>1$, each $v_i$ is connected to at least one of $v_1,\ldots,v_{i-1}$. Go over all vertices $v_1,\ldots,v_n$ in order, and color each one with the smallest available color. This algorithm also uses (at most) ...

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