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Given a graph $G$, its chromatic polynomial $\chi_G(n)$ is the function that takes a natural number $n$ to the number of proper $n$-colorings of $G$. The usual proof that this function is actually a polynomial is via deletion-contraction: if $e$ is any edge of $G$, then $\chi_G(n) = \chi_{G \setminus e}(n) - \chi_{G / e}(n)$. Some chromatic polynomials can be calculated via this same recurrence: for example cycles because deleting an edge yields a path (whose chromatic polynomial is easy) and contracting an edge yields a shorter cycle.

Are there interesting families of graphs whose chromatic polynomials have been computed by any method other than deletion-contraction?

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    $\begingroup$ Another way to prove that $\chi_G(n)$ is a polynomial is to use the inclusion-exclusion formula since $$\chi_G(n)=n^{|V(G)|}-\left|\bigcup_{uv\in E(G)}\{f\in[n]^{V(G)}:f(u)=f(v)\}\right|.$$ But I don't know if this is useful for actually calculating any chromatic polynomials. $\endgroup$ Commented Apr 2 at 22:54
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    $\begingroup$ One example is Exercise 7 on page 9 of math.mit.edu/~rstan/ec/newexer.pdf, where the computational technique is the transfer-matrix method. $\endgroup$ Commented Apr 2 at 22:56
  • $\begingroup$ For chordal graphs (en.m.wikipedia.org/wiki/Chordal_graph) you can compute their chromatic polynomials by a direct counting argument. But these are pretty trivial since they factor. $\endgroup$ Commented Apr 3 at 1:36
  • $\begingroup$ The paper pdfs.semanticscholar.org/a34b/… by Chikh and Mihoubi gives another example. $\endgroup$ Commented Apr 5 at 18:39
  • $\begingroup$ Any thoughts on the comments and answers that have been posted, Tristram? $\endgroup$ Commented Apr 6 at 3:00

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More of a remark than an answer to the question, re: proving the polynomial property other than by the contraction-deletion.

For a simple graph $G=(V,E)$, we have $$ \chi_G(n)=\sum_{c\in [n]^V}\mathbf{1}\{\text{the coloring}\ c\ \text{is proper}\} $$ $$ =\sum_{c\in [n]^V}\prod_{\{i,j\}\in E}\mathbf{1}\{c(i)\neq c(j)\} $$ $$ =\sum_{c\in [n]^V}\prod_{\{i,j\}\in E}\left(1-\mathbf{1}\{c(i)=c(j)\}\right) $$ $$ =\sum_{c\in [n]^V}\sum_{F\subseteq E}\prod_{\{i,j\}\in F}\left(-\mathbf{1}\{c(i)=c(j)\}\right) $$ $$ =\sum_{F\subseteq E}(-1)^{|F|}\sum_{c\in [n]^V}\prod_{\{i,j\}\in F}\mathbf{1}\{c(i)= c(j)\} $$ $$ = \sum_{F\subseteq E}(-1)^{|F|} n^{\gamma(V,F)} $$ where $\gamma(V,F)$ is the number of connected components of the subgraph with vertex set $V$ and edge set $F$.

This is what Alan Sokal calls the Whitney–Tutte–Fortuin–Kasteleyn representation, e.g., in this article.

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    $\begingroup$ The chromatic polynomial is essentially the characteristic polynomial of the bond lattice of the graph. (The bond lattice of the graph is the same as the geometric lattice associated to the corresponding graphic matroid.) Another way to say this is that we can compute the chromatic polynomial using Möbius inversion on the bond lattice. $\endgroup$ Commented Apr 3 at 23:07
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    $\begingroup$ The way I like to prove that $\chi_G(n)$ is a polynomial in $n$ is that it is a sum, over all partitions of the vertex set into independent sets $I_1, I_2, \ldots, I_k$, of the number of ways of coloring those independent sets with distinct colors; but this number is just $n(n-1)\cdots(n-k+1)$, which is obviously a polynomial in $n$. $\endgroup$ Commented Apr 4 at 2:43
  • $\begingroup$ Thanks, Sam. This was actually what I was thinking about when I asked the question. I've been looking at some families of graphs via Möbius inversion on the bond lattice but not getting complete formulas, and wondering if these formulas are known. $\endgroup$ Commented Apr 7 at 14:27
  • $\begingroup$ @TristramBogart I posted more details in a separate answer. But I don't understand what you mean by "not getting complete formulas." $\endgroup$ Commented Apr 10 at 19:44
  • $\begingroup$ Thanks, Sam. "Complete formulas" was sloppy; what I meant was that the Möbius inversion approach doesn't give a closed-form formula. $\endgroup$ Commented Apr 10 at 23:03
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Addition-identification is mathematically equivalent to deletion-contraction, but differs in its application. It goes, $$ \chi_G(n)=\chi_{G+e}(n)+\chi_{G/e}(n), $$ where $e$ is an edge not in $G$, $G+e$ is the result of adding $e$ to $G$, and $G/e$ is the result of identifying the endpoints of $e$. Addition-identification can be used to calculate the chromatic polynomial; for example, if $G$ is $K_{2,3}$, then letting $e$ be an edge joining the two vertices of the "$2$" in $K_{2,3}$, both terms in the formula are chordal. Also, it can be used to prove $\chi_G(n)$ is a polynomial, as repeated application results in complete graphs, for which $\chi$ is easily seen to be a polynomial.

The remark about $K_{2,3}$ applies to the family $K_{2,n}$ for all $n\ge3$.

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The following is completely standard, appearing in any textbook on the subject (e.g. see Stanley, EC1, Chapter 3 Exercise 108), but since it is not clear to me what the question asker understands I am posting it in detail.

Let $G$ be a connected, simple graph on vertex set $V$. A set partition $\pi$ of $V$ is called $G$-connected if the restriction of $G$ to each block of $\pi$ remains connected. The bond lattice $L_G$ of the graph $G$ consists of the $G$-connected partitions of $V$, ordered by refinement. So its minimal element $\hat{0}$ consists of all singleton blocks, and its maximal element $\hat{1}$ has all the vertices together in one block.

Fix an integer $n \geq 1$. For $\pi$ a $G$-connected partition of $V$, call a coloring $c\colon V \to [n]$ $\pi$-compatible if all the elements in each block of $\pi$ get the same color. Every coloring is $\hat{0}$-compatible, and notice that a coloring $c$ is proper exactly when $\pi=\hat{0}$ is the maximal element of $L_G$ for which $c$ is $\pi$-compatible. Also, the number of $\pi$-compatible colorings is clearly $n^{\#\mathrm{blocks}(\pi)}$. Hence, by Möbius inversion on $L_G$, we have that the chromatic polynomial of $G$ is $$ \chi_G(n) = \sum_{\pi \in L_G} \mu(\hat{0},\pi) \, n^{\#\mathrm{blocks}(\pi)},$$ where $\mu$ is the Möbius function of $L_G$. This means that the chromatic polynomial is (essentially) the characteristic polynomial of $L_G$.

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    $\begingroup$ A somewhat related comment is that one of the fundamental facts about the chromatic polynomial is Whitney's broken circuit theorem, which can also be formulated in a more general setting. See for example An Abstraction of Whitney's Broken Circuit Theorem by Dohmen and Trinks. $\endgroup$ Commented Apr 10 at 20:27
  • $\begingroup$ @TimothyChow It is very related. The coefficients of the characteristic polynomial of any matroid (after correcting for their alternating sign) are given by the $f$-vector of the broken circuit complex of the matroid. $\endgroup$ Commented Apr 10 at 20:41
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It sounds like you're mostly interested in closed-form formulas for infinite families of graphs, but if you're interested in efficient algorithms for computing the chromatic polynomial of graphs of moderate size, then rather than using deletion-contraction, you may be better off using methods inspired by ideas from statistical mechanics. See in particular Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions, by Marc Timme, Frank van Bussel, Denny Fliegner, and Sebastian Stolzenberg, New Journal of Physics 11 (2009), 023001, and Chromatic polynomials of random graphs, by the same authors plus Christoph Ehrlich, arXiv:1709.06209.

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