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It is famous that all integers not less than 4 enjoy this property and bit less but also famous that so does $(5+\sqrt{5})/2$. I expect that this matter was studied a lot and ask about the state of ...
Fedor Petrov's user avatar
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Let $G$ be a graph with $n$ vertices. Let $k$ be a positive integer. Assume that $G$ has no proper $\left(k-1\right)$-coloring. Then, the number of proper $k$-colorings of $G$ is at most $k! \cdot k^{...
darij grinberg's user avatar
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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 ...
Tristram Bogart's user avatar
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As mentioned in the post on Stanley's 25 positivity problems, Tatsuyuki Hikita posted a preprint on October 16, 2024 purporting to prove Problem 21, the Stanley–Stembridge conjecture about e-...
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1 answer
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Given a graph $G$ on $n$ vertices, its chromatic polynomial $P(G,x)$ is a function that gives the number of proper colorings of G using $x$ colors. When $P(G,x)$ is written using the basis $\{x, \...
ls1995's user avatar
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3 votes
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Properly colored graph (edge has color) means that any two adjacent edges have distinct colors. The edge chromatic polynomial $ech(G, k)$ gives the number of proper edge coloring of the $G$ with $k$ ...
Yuhang Bai's user avatar
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1 answer
366 views

Properly colored graph (edge has color) means that any two adjacent edges have distinct colors. For any graph with $2k-2$ edges such that it can be properly colored using $k$ colors. What is the ...
Yuhang Bai's user avatar
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Given a graph $G = (V, E)$, we can calculate its chromatic polynomial $P(G, k)$, and it has $n$ (complex) roots, also known as chromatic roots. It is a well-known fact that the sum of chromatic roots ...
Jing Guo's user avatar
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Let $\chi_G(q)$ be the chromatic polynomial of a graph $G$ with $n$ vertices. (More generally, $\chi_{\mathcal{A}}(q)$ can be the characteristic polynomial of a finite hyperplane arrangement $\mathcal{...
Richard Stanley's user avatar
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4 votes
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Suppose we have a region of the plane tiled by finitely many rectangles. We want to color the rectangles so that two rectangles have different colors if they share a part of an edge or if they share ...
Adam Chalcraft's user avatar
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1 answer
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I was playing around with the chromatic polynomial (denoted here by $\chi_G(x)$) and I have made the following conjecture. Let $(G_n)_{n \ge 1}$ be a sequence of graphs with $v(G_n) \to \infty$ ($v(...
mtsecco's user avatar
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Is there a way to obtain the chromatic polynomial of the line graph of a regular simple graph, having known the chromatic polynomial of the graph? There already exist characterizations of line graph ...
vidyarthi's user avatar
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Let $\mathcal{G}(n)$ be the isomorphism class of simple graphs of order $n$. We say two graphs in $\mathcal{G}(n)$ are chromatic equivalent if their chromatic polynomials have an equal linear ...
GA316's user avatar
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1 answer
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Consider a graph with chromatic polynomial $P(x)$ joined to a clique of order $k$ in two distinct points (joining here just means interesection of points). Then, what is the chromatic polynomial of ...
vidyarthi's user avatar
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Consider a semi-regular bipartite graph $C$ consisting of two parts $A$ (having each vertex of degree $\Delta$) and $B$ (having each vertex of degree $2$). Let its chromatic polynomial be $C(x)$. Now,...
vidyarthi's user avatar
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I am interested in the combinatorics of the linear coefficient of the chromatic polynomials. I have the following questions. What are some class of graphs for which it is possible to calculate this ...
GA316's user avatar
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1 vote
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141 views

I am struggling with this question: If I have a connected graph $G$ on $n$ vertices and $m$ edges with chromatic number $d$ then how can I give a bound(lower and upper) on its spectral radius in ...
Learnmore's user avatar
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7 votes
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In https://arxiv.org/pdf/1208.5781.pdf It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$. My ...
Matthew Levy's user avatar
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1 answer
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Let $G$ be a multigraph, i.e, there can be more than one edge between a pair of vertices. It is clear that the chromatic polynomial cannot capture these multi-edges. Because chromatic polynomial just ...
GA316's user avatar
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0 votes
1 answer
179 views

I have been working with the chromatic polynomials of circulant graphs of prime order $p$ with two distinct parameters, i.e. $P_{p,i,j}(x):= P(C_{p}(i,j),x)$ with $1 \leq i \neq j \leq \ n/2.$ In ...
Abraham G's user avatar
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1 answer
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Does anyone know the chromatic polynomial of the hyper cube graph Q4? I need this to verify that my listing of a subset of all DAG's on the 4-cube is correct. Any help greatly appreciated, JC
Jacob's user avatar
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Chromatic polynomial of a graph $G$ is an important tool in Graph theory which has been studied extensively from graph theory perspective as well as through other area of Mathematics also. Hence it is ...
GA316's user avatar
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11 votes
2 answers
718 views

Let P(n) be the set of polynomials that can occur as the chromatic polynomial of a planar map with n countries. What is known or conjectured about the growth of |P(n)|? PS: Thanks Gerry and Noam, ...
David Treumann's user avatar
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3 votes
1 answer
948 views

Given a connected graph $G$ with $n$ vertices and given set of $\{m_1,m_2,...,m_n\}$ $n$ integers, we form a new graph $G^{\wedge} $ by considering the complete graph $K_{m_i}$ for each vertex i and '...
GA316's user avatar
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6 votes
0 answers
280 views

Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then $$\...
Rebecca J. Stones's user avatar
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9 votes
1 answer
531 views

Consider the following property of a graph $G$: The graph $G$ has no independent cutset of vertices, $S$, such that the number of components of $G-S$ is more than $|S|$ (the size of $S$). (That is, ...
Gordon Royle's user avatar
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3 votes
1 answer
826 views

Suppose $c(G,u)$ is chromatic polynomial of connected simple graph $G$. We know that $|c(G,-1)|$, as Stanley proved, is the total number of directed graph on $G$, without any cycle. Also, we know ...
Shahrooz's user avatar
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1 vote
1 answer
170 views

It is well-known that the coefficients of a chromatic polynomial alternate in sign. But is it possible for a chromatic polynomial to have a factor (over $\mathbb{Q}$) with coefficients which do not ...
Adam 's user avatar
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5 votes
2 answers
1k views

Let $p(x)$ be the chromatic polynomial of a special graph. Performing a certain type of operation on the graph changes $p$ by shifting it and adding a constant, say to: $q(x)=p(x+a) + b$. I have ...
Adam 's user avatar
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11 votes
2 answers
920 views

Does anyone know more examples of two nonisomorphic connected graphs with the same chromatic symmetric function? The only pair I know is the one in Stanley's paper on c.s.f.'s [http://math.mit.edu/~...
Jeremy Martin's user avatar
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13 votes
0 answers
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I would like to find the chromatic polynomial χ for the n by m rook's graph Gn,m for as many values of n and m possible. The rooks graph is also (a) the line graph of the complete bipartite graph ...
Douglas S. Stones's user avatar
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