This paper describes an improvement in the upper bound for the magnitude of a coefficient of a term in the chromatic polynomial of a general graph. If $a_r$ is the coefficient of the $q^r$ term in the chromatic polynomial $P(G,q)$, where $q$ is the number of colors, then we find $a_r \le {e \choose v-r} - {e-g+2 \choose v-r-g+2} + {e-k_g-g+2 \choose v-r-g+2} - \sum _{n=1}^{k_g-\ell_g}\sum_{m=1}^{\ell_g-1} {e-g+1-n-m \choose v-r-g} - δ_{g,3}\sum_{n=1}^{k_g+\ell_{g+1}^*-\ell_g} {e-\ell_g-g+1-n \choose v-r-g}$, where $k_g$ is the number of circuits of length $g$ and $\ell_g$ and $\ell_{g+1}^*$ are certain numbers defined in the text.
This paper describes how many known graph polynomials arise from the coefficients of chromatic symmetric function expansions in different bases, and studies a new polynomial arising by expanding over a basis given by chromatic symmetric functions of trees.
Using the definition of colouring of $2$-edge-coloured graphs derived from $2$-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to $2$-edge-coloured graphs. We find closed forms for the first three coefficients of this polynomial that generalize the known results for the chromatic polynomial of a graph. We classify those $2$-edge-coloured graphs that have a chromatic polynomial equal to the chromatic polynomial of the underlying graph, when every vertex is incident to edges of both colours. Finally, we examine the behaviour of the roots of this polynomial, hig…
In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph $G=(V,E)$, the number of totally cyclic orientations of $G$ is equal to the value of $|P(H,-1)|$, where $P(H,λ)$ is the chromatic polynomial of a hypergraph $H$ which is constructed from $G$. Finally we show that the multiplicity of root "$0$" of $P(H,λ)$ may be at least $2$ for…
In this paper, we prove that $(-\infty, 0)$ is a zero-free interval for chromatic polynomials of a family ${\cal L}_0$ of hypergraphs and $(0, 1)$ is a zero-free interval for chromatic polynomials of a subfamily ${\cal L}_0'$ of ${\cal L}_0$ of hypergraphs. These results extend known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.
We use a polynomial decomposition result by Stapledon to show that the numerator polynomial of the Ehrhart series of an open polytope is the difference of two symmetric polynomials with nonnegative integer coefficients. We obtain a related decomposition for order polytopes and for the numerator polynomial of the corresponding series for chromatic polynomials. The nonnegativity of the coefficients in such decompositions provide inequalities satisfied by the coefficients of chromatic polynomials for any simple graph.
The chromatic polynomials are studied by several authors and have important applications in different frameworks, specially, in graph theory and enumerative combinatorics. The aim of this work is to establish some properties of the coefficients of the chromatic polynomial of a graph. Three applications on restricted Stirling numbers of the second kind are given.
We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and power sum symmetric functions to provide Schur and power sum formulas for the integral form Macdonald polynomials. Since the (integral form) Jack polynomials are a specialization of integral form Macdonald polynomials, we obtain analogous formulas for Jack polynomials as corollaries.
The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree chromatic polynomial of a tree.
Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue of the chromatic quasisymmetric function called the \emph{chromatic nonsymmetric polynomial} of a Dyck graph. We give a positive expansion for this polynomial in the basis of fundamental slide polynomials using recent work of Assaf-Bergeron on flagged $(P,ρ)$-partitions. We then derive the known expansion for the chromatic quasisymmetric function of Dyck graphs in terms of Gessel's fundamental basis by taking a backstable limit of our expansion.
In this paper, we consider multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously investigated for graphs by Averbouch, Godlin, and Makowsky. The multivariate chromatic polynomial is an equivalent polynomial defined in terms of colorings, and generalizes the coboundary polynomial of Crapo, and the bivariate chromatic polynomial of Dohmen, Pönitz and Tittman. We show that specializations of these new polynomials recover polynomials which en…
Let G be a graph, and let $χ$G be its chromatic polynomial. For any non-negative integers i, j, we give an interpretation for the evaluation $χ$ (i) G (--j) in terms of acyclic orientations. This recovers the classical interpretations due to Stanley and to Green and Zaslavsky respectively in the cases i = 0 and j = 0. We also give symmetric function refinements of our interpretations, and some extensions. The proofs use heap theory in the spirit of a 1999 paper of Gessel.
Problem of finding an optimal upper bound for a chromatic no. of 3K1-free graphs is still open and pretty hard. Here we find a tight chromatic upper bound for {3K1, C5}-free graphs. We prove that if G is {3K1, C5}-free, then the chromatic no. <= (3ω-1)/2 where ω is the size of a maximum clique in G and show with examples that the bound is tight.
Chromatic polynomials have been studied extensively, giving us results such as the Fundamental Reduction Theorem and closed formulas for the chromatic polynomials of common classes of graphs. Though, none of those extend to the context of fractional colorings. We thus present closed formulas for the "fractional" chromatic polynomial - a function that counts the number of distinct $b$-fold $λ$-colorings on a given graph - of complete graphs, trees and forests, as well as a generalized form of the Fundamental Reduction Theorem.
Let $G = (V,E)$ be a finite, simple, connected graph with chromatic polynomial $P_G(q)$. Sokal \cite{sokal} proved that the roots of the chromatic polynomial of $G$ are bounded in absolute value by $KD$ where, $D$ is the maximum degree of the graph and $7< K < 8$ is a constant. In this paper we generalize this result to uniform hypergraphs. To prove our results we will use the theory of the bounded exponential type graph polynomials.
Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization of the Stanley's one. Our goal is to extend Stanley's chromatic polynomial to embedded graphs. In contrast to well-known extensions of, say, the Tutte polynomial from abstract to embedded graphs [5], we do not treat an embedded graph as an abstract graph endowed with additional information about the embedding. Instead, we consider the binary delta-matroid a…
Problem of finding an optimal upper bound for the chromatic no. of 3K1-free graphs is still open and pretty hard. It was proved by Choudum et al that an upper bound on the chromatic no. of {3K1, K1+C4}-free graphs, is 2ω. We improve this by proving that if G is {3K1, K1+C4}-free, then its chromatic no. is less than or equal to 3ω divided by 2, where ω is the size of a maximum clique in G. Also we give examples to show that this bound is tight.
Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to give certain conditions under which two of the graphs have chromatic polynomials with the same splitting field. Finally, we use a subfamily of bicliques to prove the cubic case of the $α+n$ conjecture, by showing that for any cubic integer $α$, there is a natural number $n$ such that $α+n$ is a chromatic roo…
We study "holomorphic quadratic differentials" on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial $χ$ of a graph $G$, at negative integer values, can be evaluated as the degree of a certain rational mapping, arising from the defining equations for a holomorphic quadratic differential. This allows us to give an explicit integral expression for $χ(-k)$.
Given a group $G$ of automorphisms of a graph $Γ$, the orbital chromatic polynomial $OP_{Γ,G}(x)$ is the polynomial whose value at a positive integer $k$ is the number of orbits of $G$ on proper $k$-colorings of $Γ.$ In \cite{Cameron}, Cameron et. al. explore the roots of orbital chromatic polynomials, and in particular prove that orbital chromatic roots are dense in $\mathbb{R}$, extending Thomassen's famous result (see \cite{Thomassen}) that chromatic roots are dense in $[\frac{32}{27},\infty)$. Cameron et al \cite{Cameron} further conjectured that the real roots of the orbital chromatic pol…
It is proved that if $G$ is a graph containing a spanning tree with at most three leaves, then the chromatic polynomial of $G$ has no roots in the interval $(1,t_1]$, where $t_1 \approx 1.2904$ is the smallest real root of the polynomial $(t-2)^6 +4(t-1)^2(t-2)^3 -(t-1)^4$. We also construct a family of graphs containing such spanning trees with chromatic roots converging to $t_1$ from above. We employ the Whitney $2$-switch operation to manage the analysis of an infinite class of chromatic polynomials.
The chromatic polynomial is a well studied object in graph theory. There are many results and conjectures about the log-concavity of the chromatic polynomial and other polynomials related to it. The location of the roots of these polynomials has also been well studied. One famous result due to A. Sokal and C. Borgs provides a bound on the absolute value of the roots of the chromatic polynomial in terms of the highest degree of the graph. We use this result to prove a modification of a log-concavity conjecture due to F. Brenti. The original conjecture of Brenti was that the chromatic polynomial…
Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs and that they could be told apart by their chromatic polynomials, by showing that the latter give distinct results when evaluated at 3. He conjectured that the six different signed Petersen graphs also have distinct zero-free chromatic polynomials, and that both types of chromatic polynomials have distinct evaluations at \emph{any} positive integer. We developed and executed a computer program (running in {\tt SAGE}) that efficiently determines the number of proper $k$-colorings for a giv…
We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at $-1$ up to sign. Motivated by this link between acyclic orientations and the chromatic polynomial, we develop "acyclic orientation" analogues of theorems concerning the chromatic polynomial of Birkhoff, Whitney, and Greene-Zaslavsky. As an application, we provide a new proof for Stanley's sink theorem for chromatic symmetric functions $X_G$. This theorem gives a r…
For $m \geq 3$ and $n \geq 1$, the $m$-cycle book graph $B(m,n)$ consists of $n$ copies of the cycle $C_m$ with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed $B(m,n)$ is $n+1$, and (b) the chromatic number of a signed $B(m,n)$ is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs.
In 2015, Matthias Beck and his team developed a computer program in SAGE which efficiently determines the number of signed proper $k$-colorings for a given signed graph. In this article, we determine the number of different signatures on Book graph up to switching isomorphisms. We also find a recursive formula of the signed chromatic polynomials of signed Book graphs.
In recent years we have worked on a project involving poset topology, various analogues of Eulerian polynomials, and a refinement of Richard Stanley's chromatic symmetric function. Here we discuss how Stanley's ideas and results have influenced and inspired our own work.
We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as vertical strip --- in particular, unicellular LLT polynomials. We show that there are parallel phenomena regarding $e$-positivity of these two families of polynomials. In particular, we give several examples where the LLT polynomials behave like a "mirror image" of the chromatic quasisymmetric counterpart. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials. This circular extensions of LLT polynomials…
An acyclic coloring of a digraph as defined by Neumann-Lara is a vertex-coloring such that no monochromatic directed cycles occur. Counting the number of such colorings with $k$ colors can be done by counting so-called Neumann-Lara-coflows (NL-coflows), which build a polynomial in $k$. We will present a representation of this polynomial using totally cyclic subdigraphs, which form a graded poset $Q$. Furthermore we will decompose our NL-coflow polynomial, which becomes the chromatic polynomial of a digraph by multiplication with the number of colors to the number of components, examining the s…
We establish closed-form expansions for the number of colorings of a path or cycle on n vertices with colors from 1,...,x such that adjacent vertices are colored differently or with colors from y+1,...x.
We study Dohmen--Pönitz--Tittmann's bivariate chromatic polynomial $c_Γ(k,l)$ which counts all $(k+l)$-colorings of a graph $Γ$ such that adjacent vertices get different colors if they are $\le k$. Our first contribution is an extension of $c_Γ(k,l)$ to signed graphs, for which we obtain an inclusion--exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for $c_Γ(k,l)$ and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynom…
The multi-fold chromatic number of the plane $χ_m$ is the smallest number of colors $k$, sufficient to color each point of the Euclidean plane in exactly $m$ colors, so that for any pair of points at a unit distance from each other, two corresponding $m$-subsets of $k$-set do not contain any common color. We consider upper bounds for $m$-fold chromatic numbers of the plane. Our main result is that for any $m$ the inequality $χ_m<(1+2/\sqrt3)^2\cdot m+3.501$ holds.
Stanley in his paper [Stanley, Richard P.: Acyclic orientations of graphs In: Discrete Mathematics 5 (1973), Nr. 2, S. 171-178.] provided interpretations of the chromatic polynomial when it is substituted with negative integers. Greene and Zaslavsky interpreted the coefficients of the chromatic polynomial in [Greene, Curtis ; Zaslavsky, Thomas: On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. In: Transactions of the American Mathematical Society 280 (1983), jan, Nr. 1, S. 97-97.]. We shall develop an invo…
In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph $H$, the maximum number $k$ for which there is a balanced rainbow-free $k$-coloring of $H$ is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane $\mathrm{PG}(2,q)$ for all $q$. In addition, we det…
In this paper, we propose an algebraic approach to determine whether two non-isomorphic caterpillar trees can have the same symmetric function generalization of the chromatic polynomial. On the set of all composition on integers, we introduce: An operation, which we call composition product; and a combinatorial polynomial, which we call the composition-lattice polynomial or L-polynomial, that mimics the weighted graph polynomial of Noble and Welsh. We prove a unique irreducible factorization theorem and establish a connection between the L-polynomial of a composition and its irreducible factor…
The main result of this paper is the introduction of marked graphs and the marked graph polynomials ($M$-polynomial) associated with them. These polynomials can be defined via a deletion-contraction operation. These polynomials are a generalization of the $W$-polynomial introduced by Noble and Welsh and a specialization of the $\mathbf{V}$-polynomial introduced by Ellis-Monaghan and Moffatt. In addition, we describe an important specialization of the $M$-polynomial which we call the $D$-polynomial. Furthermore, we give an efficient algorithm for computing the chromatic symmetric function of a …
A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a $P_5$-free graph with clique number $ω\ge 3$ has chromatic number at most $ω^{\log_2(ω)}$. The best previous result was an exponential upper bound $(5/27)3^ω$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdos-Hajnal conjecture holds for $P_5$, which is the smallest open case. Thus there is great interest in whether there is a polynomial bound for $P_5$-free graphs, and our result is an attempt to approach that.
A famous and wide-open problem, going back to at least the early 1970's, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute such inequalities when a chromatic polynomial $χ_G(n) = χ^*_0 \binom {n+d} d + χ^*_1 \binom {n+d-1} d + \dots + χ^*_d \binom n d$ is written in terms of a binomial-coefficient basis. For example, we show that $χ^*_{ j } \le χ^*_{ d-j }$, for $0 \le j \le \frac{ d }{ 2 }$. Similar results hold for flow and tension polyn…
In this work, we generalize and utilize the linear relations of LLT polynomials introduced by Lee \cite{Lee}. By using the fact that the chromatic quasisymmetric functions and the unicellular LLT polynomials are related via plethystic substitution and thus they satisfy the same linear relations, we can apply the linear relations to both sets of functions. As a result, in the chromatic quasisymmetric function side, we find a class of $e$-positive graphs, called \emph{melting lollipop graphs}, and explicitly prove the $e$-unimodality. In the unicellular LLT side, we obtain Schur expansion formul…
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 $H$, there is a function $f$ such that if $G$ does not contain $H$ as an induced subgraph then $χ(G)\le f(ω(G))$ (where $χ, ω$ are the chromatic number and the clique number of $G$). Here we prove that $f$ can be chosen to be a polynomial.
We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $χ_G(k,l)$ that counts the number of signed colorings using colors $0,\pm1,\dots,\pm k$ along with $l-1$ symmetric colors $0_1,\dots,0_{l-1}$. We show that the evaluation of the bivariate chromatic polynomial $|χ_G(-1,2)|$ is equal to the number of acyclic orientations of the signed graph modulo the equivalence relation generated by swapping sources and sinks. We present three proofs of this fact, a proof using toric hyperplan…
A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we present a polynomial time algorithm that finds an optimum star edge coloring for every tree. We also provide some tight bounds on the star chromatic index of trees with diameter at most four, and using these bounds we find a formula for the star chromatic index of certain families of trees.
If $G$ is a $k$-chromatic graph of order $n$ then it is known that the chromatic polynomial of $G$, $π(G,x)$, is at most $x(x-1)\cdots (x-(k-1))x^{n-k} = (x)_{\downarrow k}x^{n-k}$ for every $x\in \mathbb{N}$. We improve here this bound by showing that \[ π(G,x) \leq (x)_{\downarrow k} (x-1)^{Δ(G)-k+1} x^{n-1-Δ(G)}\] for every $x\in \mathbb{N},$ where $Δ(G)$ is the maximum degree of $G$. Secondly, we show that if $G$ is a connected $k$-chromatic graph of order $n$ where $k\geq 4$ then $π(G,x)$ is at most $(x)_{\downarrow k}(x-1)^{n-k}$ for every real $x\geq n-2+\left( {n \choose 2} -{k \choose…
Let $G=(V,E)$ be a simple graph with $V=\{1,2,\cdots,n\}$ and $χ(G,x)$ be its chromatic polynomial. For an ordering $π=(v_1,v_2,\cdots,v_n)$ of elements of $V$, let $δ_G(π)$ be the number of $i$'s, where $1\le i\le n-1$, with either $v_i<v_{i+1}$ or $v_iv_{i+1}\in E$. Let ${\cal W}(G)$ be the set of subsets $\{a,b,c\}$ of $V$, where $a<b<c$, which induces a subgraph with $ac$ as its only edge. We show that ${\cal W}(G)=\emptyset$ if and only if $(-1)^nχ(G,-x)=\sum_π {x+δ_G(π)\choose n}$, where the sum runs over all $n!$ orderings $π$ of $V$. To prove this result, we establish an analogous resu…
We prove that for every path H, and every integer d, there is a polynomial f such that every graph G with chromatic number greater than f(t) either contains H as an induced subgraph, or contains as a subgraph the complete d-partite graph with parts of cardinality t. For t = 1 and general d this is a classical theorem of Gyárfás, and for d = 2 and general t this is a theorem of Bonamy et al.
The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of $X_G$ to $χ_G(λ)$, the chromatic polynomial, we also define a generalization $χ^k_G(λ)$ and show that evaluations of this polynomial for negative values generalize a theorem of Stanley relating acyclic orientations to the chromatic polynomial.
A \emph{mixed graph} is a graph with directed edges, called arcs, and undirected edges. A $k$-coloring of the vertices is proper if colors from ${1,2,...,k}$ are assigned to each vertex such that $u$ and $v$ have different colors if $uv$ is an edge, and the color of $u$ is less than or equal to (resp. strictly less than) the color of $v$ if $uv$ is an arc. The weak (resp. strong) chromatic polynomial of a mixed graph counts the number of proper $k$-colorings. Using order polynomials of partially ordered sets, we establish a reciprocity theorem for weak chromatic polynomials giving interpretati…
Let H be a tree. It was proved by Rodl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$ as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing that such graphs have bounded degeneracy. Here we give a further strengthening, proving that for every tree H, the degeneracy is at most polynomial in t. This answers a question of Bonamy, Pilipczuk, Rzazewski, Thomasse and Walczak.
