Chromatic polynomial
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Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree Open
Let be a tree. It was proved by Rödl that graphs that do not contain as an induced subgraph, and do not contain the complete bipartite graph as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing tha…
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On chromatic polynomial of certain families of dendrimer graphs Open
Let \\(G\\) be a simple graph with vertex set \\(V(G)\\) and edge set \\(E(G)\\). A mapping \\(g:V (G)\\rightarrow\\{1,2,...t\\}\\) is called \\(t\\)-coloring if for every edge \\(e = (u, v)\\), we have \\(g(u) \\neq g(v)\\). The chromatic…
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TUTTE POLYNOMIALS FOR DIRECTED GRAPHS Open
The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name the B-polynomial. The B-polynomial has three variables, but when spe…
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Polynomial bounds for chromatic number. III. Excluding a double star Open
A “double star” is a tree with two internal vertices. It is known that the Gyárfás–Sumner conjecture holds for double stars, that is, for every double star , there is a function such that if does not contain as an induced subgraph then (wh…
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Symmetric Chromatic Polynomial of Trees Open
In a 1995 paper Richard Stanley defined $X_G$, the symmetric chromatic polynomial of a Graph $G=(V,E)$. He then conjectured that $X_G$ distinguishes trees; a conjecture which still remains open. $X_G$ can be represented as a certain collec…
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A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function Open
This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of XB and show that this function admits a deletion-contraction…
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Linear Bound in Terms of Maxmaxflow for the Chromatic Roots of Series-Parallel Graphs Open
We prove that the (real or complex) chromatic roots of a series-parallel graph with maxmaxflow Lambda lie in the disc |q-1| < (Lambda-1)/log 2. More generally, the same bound holds for the (real or complex) roots of the multivariate Tutte …
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A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function Open
This paper has two main parts. First, we consider the Tutte symmetric function $XB$, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of $XB$ and show that this function admits a deletion-contrac…
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Counting faces of graphical zonotopes Open
It is a classical fact that the number of vertices of the graphical zonotope ZΓ is equal to the number of acyclic orientations of a graph Γ . We show that the f-polynomial of ZΓ is obtained as the principal specialization of the q-analog o…
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The Chromatic Number of a Signed Graph Open
In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the colour…
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Chromatic zeros on hierarchical lattices and equidistribution on parameter space Open
Associated to any finite simple graph \Gamma is the chromatic polynomial \mathcal{P}_\Gamma(q) whose complex zeros are called the chromatic zeros of \Gamma . A hierarchical lattice is a sequence of finite simple graphs \{\Gamma_n\}_{n=0}^\…
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Types of embedded graphs and their Tutte polynomials Open
We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graph…
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A convolution formula for Tutte polynomials of arithmetic matroids and other combinatorial structures Open
In this note we generalize the convolution formula for the Tutte polynomial of Kook-Reiner-Stanton and Etienne-Las Vergnas to a more general setting that includes both arithmetic matroids and delta-matroids. As corollaries, we obtain new p…
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Gadgets and Anti-Gadgets Leading to a Complexity Dichotomy Open
We introduce an idea called anti-gadgets for reductions in complexity theory. These anti-gadgets are presented as graph fragments, but their effect is equivalent to erasing the presence of other graph fragments, as if we had managed to inc…
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The Cycle Polynomial of a Permutation Group Open
The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles:\[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $\Omega$. …
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Algebraic aspects of the chromatic polynomial Open
The chromatic polynomial P(G, λ) gives the number of proper colourings of a graph G in at most λ colours. Although there has been considerable interest in the chromatic polynomial, there has been little research into its algebraic theory. …
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The interlace polynomial of binary delta-matroids and link invariants Open
In this work, we study the interlace polynomial as a generalization of a graph invariant to delta-matroids. We prove that the interlace polynomial satisfies the four-term relation for delta-matroids and determines thus a finite type invari…
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Evaluations of Tutte polynomials of regular graphs Open
Let TG(x,y) be the Tutte polynomial of a graph G. In this paper we show that if (Gn)n is a sequence of d-regular graphs with girth g(Gn)→∞, then for x≥1 and 0≤y≤1 we havelimn→∞TGn(x,y)1/v(Gn)=td(x,y), wheretd(x,y)={(d−1)((d−1)2(d−1)2−x)d/…
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Planar graphs and Stanley's Chromatic Functions Open
This article is dedicated to the study of positivity phenomena for the chromatic symmetric function of a graph with respect to various bases of symmetric functions. We give a new proof of Gasharov's theorem on the Schur-positivity of the c…
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A Cvetković-type Theorem for coloring of digraphs Open
In 1972, Cvetković proved that if G is an n-vertex simple graph with the chromatic number k, then its spectral radius is at most the spectral radius of the n-vertex balanced complete k-partite graph. In this paper, we analyze the character…
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A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials Open
In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial \(G_{\boldsymbol\lambda}(\boldsymbol x;q)\) in some special cases. We associate a weighted grap…
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Hopf algebras and Tutte polynomials Open
By considering Tutte polynomials of Hopf algebras, we show how a Tutte\npolynomial can be canonically associated with combinatorial objects that have\nsome notions of deletion and contraction. We show that several graph\npolynomials from t…
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Structure of the flow and Yamada polynomials of cubic graphs Open
We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at …
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Tutte polynomials and random-cluster models in Bernoulli cell complexes Open
This paper studies Bernoulli cell complexes from the perspective of persistent homology, Tutte polynomials, and random-cluster models. Following the previous work [9], we first show the asymptotic order of the expected lifetime sum of the …
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Colorings and flows on CW complexes, Tutte quasi-polynomials and arithmetic matroids Open
In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials encompasses invariants defined for CW co…
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Algebraic Properties of Chromatic Roots Open
A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as a…
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A generalization of the Tutte polynomials Open
In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariant…
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On P-unique hypergraphs Open
We study hypergraphs which are uniquely determined by their chromatic, independence and matching polynomials. B. Bollobás, L. Pebody and O. Riordan (2000) conjectured (BPR-conjecture) that almost all graphs are uniquely determined by their…
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Gröbner Bases Techniques for an $S$-Packing $k$-Coloring of a Graph Open
In this paper, polynomial ideal theory is used to deal with the problem of the $S$-packing coloring of a finite undirected and unweighted graph by introducing a family of polynomials encoding the problem. A method to find the $S$-packing c…
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Bears with Hats and Independence Polynomials Open
Consider the following hat guessing game. A bear sits on each vertex of a graph $G$, and a demon puts on each bear a hat colored by one of $h$ colors. Each bear sees only the hat colors of his neighbors. Based on this information only, eac…