I am looking for examples of problems or proofs in mathematics which have a equivalent representation in terms of graphs, which makes solving the problem easier.
For example, the problem of finding how many inversions are required to sort an array can be solved by representing the problem as a graph.
Here's a problem from the AoPS text
$\qquad$Patrick -- Intermediate Counting and Probability (2007)
(original source was USAMO) which does not immediately appear to be Graph Theory related, but which is solved (in the text on page 345) very nicely via Graph Theory.
During a certain lecture, each of 5 mathematicians fell asleep exactly twice. For each pair of these mathematicians, there was some moment when both were sleeping simultaneously. Prove that, at some moment, some three of them were sleeping simultaneously.
Final exam scheduling. Find the minimum number of time slots needed so that each student's exams appear in different slots. Define a node for each exam and an edge between each pair of exams that share a student. You want the chromatic number of this graph. Each color corresponds to a time slot.
Several examples are given in this my answer.
I can add to them the following story. Last year I was a tutor for a block-course of an algorithmic graph theory in Ukrainian Catholic University. I heard as one of my students tells that his boss sometimes poses problems for the staff. The current problem was the following.
A magician with his apprentice are preparing the following trick. First, a volunteer writes a sequence of $n$ digits on a blackboard. Then the apprentice erases two neighboring digits. Finally, the magician enters the room and using the mystic power of his mind recovers the erased digits (and their order). Unluckily, yesterday the magician too jolly celebrated three hundred year anniversary of his master, so today his mind powers are restricted to memorization and calculation. Find a minimal number $n$ for which the magician can provide the apprentice with instructions guaranteeing a success of the trick.
Since the student told that his boss fires those who is unable to solve the problems, I felt obliged to save him. I solved the problem, greatly promoting the real usefulness of our graph theory course for practical needs. Here is a strong hint.
Apply Hall's marriage theorem.
This spring I was looking new issues of Russian mathematical journal “Квант” (“Quantum”) and found that this problem is from All-Russian mathematical olympiad in 2007. The solution was the same as mine, see Shvetsova’s article here, at page 28 (in Russian, of course).
Here's a very common sort of problem that has applications in lots of areas from the theory of codes to dynamical system.
How many strings of $0$s and $1$s are there of length $n$ which avoid the substring $00$?
Now there are various methods for solving this using induction and some simple combinatorics, but this can get quite arduous when we replace the substring $00$ with something more complicated like $00100$ or similar. So this method isn't easily generalisable.
The better approach is to encode the problem as one in graph theory.
Let us build an associated directed graph $G = (V,E)$. Label the vertices $V = \{01, 10, 11\}$ (all of the allowed length-$2$ substrings) and then attach a directed edge from $v_1$ to $v_2$ if the last symbol of $v_1$ is equal to the first symbol of $v_2$. So we have the set of edges
$$E = \{(01,10), (01,11), (10,01) (11,10), (11,11)\}.$$
The question can now be rephrased as follows.
How many directed paths of length $n$ are there in the graph $G$?
This is a very typical graph theory problem and is generally computed by taking powers of the associated transition matrix $M$. That is, the number of directed paths of length $n$ in $G$ is given by the sum of the entries of $M^n$, or rather $$ \begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix}^TM^n \begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix}^T \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \\ \end{bmatrix}^n \begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix} $$ and this can be solved exactly by diagonalising $M$. (Try doing this for yourself and notice the surprising appearance of the golden ratio $\phi = \frac{1}{2}(1 + \sqrt{5})$.)
This method works more generally, and so we can build a corresponding graph for any finite set of avoiding string $s_1, \ldots, s_k$, not just $s=00$.
This kind of approach forms the basis for how dynamicists (and others) study what are called 'subshifts of finite type' or 'topological Markov chains'.
The Witten conjecture on the intersection numbers of stable classes on the moduli space of curves was proved by Kontsevich by considering the graphs embedded on genus $g$ surfaces with $n$ marked points. However, this is far too complex to even attempt to explain here, but here is something in a similar vein:
Problem: Let $P$ and $Q$ be polynomials over $\mathbb C$. What is the smallest possible degree of the polynomial $P^3 - Q^2$, in terms of degrees of $P$ and $Q$?
Answer: for monic $P$ and $Q$ with $P^3\neq Q^2$ and $\deg P = 2m$ and $\deg Q = 3m$ we have $\deg(P^3-Q^2)\geq m + 1$, with the equality attained for infinitely many $m$.
This is the Davenport-Stothers-Zannier bound, conjectured by Davenport in 1965, the inequality proven by Davenport in 1965, the sharpness by Stothers in 1981, and generalised by Zannier in 1995.
This problem can be reformulated in terms of plane graphs:
Reformulation: is there, for every $m$, a plane graph such that it has $3m$ edges, $2m$ vertices of degree 3, and such that all but one face are of degree 1?
Answer: yes, proof: For inductive basis, take $m=1$. Draw the tree with 2 vertices and 1 edge, and add a loop to every vertex (a loop is counted twice for the purposes of degree). Inductive step: delete a single loop from a vertex $v$ and add two leafs to $v$. Now add a loop to each new vertex.
Why does this prove the bound? Plane graphs with $V$ vertices, $E$ edges and $F$ faces are in a 1-1 correspondence with holomorphic maps $f\colon\mathbb C\to\mathbb C$ over $\overline{\mathbb Q}$ of degree $E$ such that $f^{-1}(0)$ has $V$ elements, $f^{-1}(1)$ has $E$ elements, and $f^{-1}(\infty)$ has $F-1$ elements (plus one element at infinity), with ramification indices of points in the preimages $f^{-1}(0)$ and $f^{-1}(\infty)$ corresponding to the degrees of vertices and faces (this correspondence is up to orientation preserving homeomorphisms on the graph side, and composing with a Möbius transformation on the holomorphic side), i.e. maps $f=g/h$ such that $g$ and $h$ are polynomials where the roots of $g$ correspond to the vertices of $G$, the roots of $g-h$ to the edges of $G$, and the roots of $h$ to the faces of $G$.
This is a special case of the so-called Grothendieck-Belyi correspondence, and the fundamental result in the Theory of Dessins d'Enfants, which states that the pairs $(X,f)$, where $X$ is a compact Riemann surface defined over the algebraic numbers, and $f\colon X\to\hat{\mathbb{C}}$ is a holomorphic map to the Riemann sphere $\hat{\mathbb{C}}$, ramified over $\{0,1,\infty\}$, are in a 1-1 correspondence with graphs cellularly embedded on compact surfaces. This correspondence gives a dictionary between problems about algebraic curves and problems about embedded graphs.
Now suppose $P^3(x)-Q^2(x)=R(x)$. If $f(x)=\frac{P^3(x)}{R(x)}$, then $f(x)-1=\frac{Q^2(x)}{R(x)}$. If $P$ has degree $2m$ and no multiple roots, then the zeros of $P^3$ correspond to $2m$ vertices of degree 3. If $Q$ has degree $3m$ and no multiple roots, then the zeros of $Q^2$ corresponds to $3m$ edges. By the Euler-Poincare formula, a plane graph with $2m$ vertices and $3m$ edges has $m+2$ faces. Therefore, $R$ must have $m+1$ distinct zeros, since once face is attained for $x=\infty$ (here $\infty$ is understood as "the north pole" on the Riemann sphere).
The 1981 paper of Stothers which first completed the proof of Davenport's conjecture has approx. 20 pages. But, as you can see, when the dictionary between holomorphic maps and plane graphs is established (which at present time is fairly well known, and covered by several books through an elementary approach), the proof is reduced to half a page.
Reference: Lando, Zvonkin - Graphs on Surfaces and their Applications, with appendix by D. Zagier, section 2.5.
