Mekkia Kouider, Manouchehr Zaker
In this paper we obtain some upper bounds for $b$-chromatic number of $K_{1,t}$ -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs or biclique number for bipartite graphs. We show that all the bounds are tight.
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On local antimagic chromatic number of the join of two special families of graphs -- II
本文研究局部反魔幻色数，证明了1-正则图与零图的联图具有局部反魔幻色数3，并由此构造出多类具有该色数3的二部图和三部图族。
Given a minor-closed class of graphs $\mathcal{G}$, what is the infimum of the non-trivial roots of the chromatic polynomial of $G \in \mathcal{G}$? When $\mathcal{G}$ is the class of all graphs, the answer is known to be $32/27$. We answer this question exactly for three minor-closed classes of graphs. Furthermore, we conjecture precisely when the value is larger than $32/27$.
We consider class of integral sum graphs $H^{-i,s}_{m,j}$ subject to the conditions $-i<0<s$, $1\leq m < i$ and $1\leq j < s$ for all $i,s, m,j\in \mathbb{N}$. We apply edge-sum coloring and edge coloring on $H^{-i,s}_{m,j}$. Since the graphs fully depend on $i$ and $s$, therefore it is not easy to derive the theoretical as well as numerical results for all values of $i$ and $s$. Here, we derive the general formula for computing the minimum number of independent color classes. We compute the edge chromatic as well as edge-sum chromatic number of $H^{-i,s}_{m,j}$ corresponding to different valu…
On local antimagic chromatic numbers of the join of two special families of graphs
本文通过构造$(2m+1) \times (2k+1)$矩阵，完全确定了零图与奇数分支1-正则图的联图的局部反魔幻色数，并由此得到无穷多个局部反魔幻色数为3的三部图。
Let $Γ$ be the graph whose vertices are the chambers of the finite projective space $PG(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\ge 43$ we determine the largest independent set of $Γ$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\ge 47$, this information is then used to show that $Γ$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove simi…
New Families of tripartite graphs with local antimagic chromatic number 3
该研究聚焦图论中的局部反魔幻标号问题，利用固定大小的矩阵构造出多族无穷多个局部反魔幻色数为3的三部图，丰富了相关图族的研究成果。
On the exact quantum chromatic number of generalized Johnson graphs
本文研究广义约翰逊图的量子色数，通过构造正交表示、结合特征值分析确定两类无穷族的精确量子色数，发现其经典色数随n指数增长、量子色数线性增长，存在指数级分离。
On Grundy and b-chromatic number of some families of graphs: a comparative study
本文比较研究图的 Grundy 数与 b-色数，构造二者背离的图序列，并证明若干图族中 Grundy 数可由 b-色数的函数控制，从而验证相关猜想。
The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple families of trees, including trees of radius two, some special trees of radius three, and subdivided stars. These trees all have the property that their vertices of degree more than two are clustered quite closely together. In this paper, we prove the conjecture for two families of trees which do not have this restriction. As special cases, these families contain …
We exhibit infinite families of planar graphs with real chromatic roots arbitrarily close to 4, thus resolving a long-standing conjecture in the affirmative.
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…
The Grundy number of a graph $G$ is the maximum number of colors used by the First-Fit coloring of $G$ and is denoted by $Γ(G)$. Similarly, the ${\rm b}$-chromatic number ${\rm{b}}(G)$ of $G$ expresses the worst case behavior of another well-known coloring procedure i.e. color-dominating coloring of $G$. We obtain some families of graphs $\mathcal{F}$ for which there exists a function $f(x)$ such that $Γ(G)\leq f({\rm{b}}(G))$, for each graph $G$ from the family. Call any such family $(Γ,b)$-bounded family. We conjecture that the family of ${\rm b}$-monotone graphs is $(Γ,b)$-bounded and valid…
The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$. The least value $k$ for which $G$ admits such labeling is called \emph{additive chromatic number} of $G$. This problem was first presented by Czerwiński, Grytczuk and Żelazny who also proposed a conjecture that for every graph $G$, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete …
Upper bounds for the list-distinguishing chromatic number
本文针对简单有限连通图，证明了Brooks定理在列表可区分色数上的类比，并利用着色数和列表色数给出两个上界，同时确定了书图等多种图族的列表可区分色数。
Cover numbers by graph families bounded by certain graph parameters
研究由分数色数和局部色数界定的图族的覆盖数问题，证明分数色数至多为$\beta$时无类似公式，给出上下界，还研究了特定图的覆盖数，得到局部色数至多为$k$时关于$\psi$的上界和关于$\omega$的下界。
Chromatic polynomials of signed graphs and dominating-vertex deletion formulae
本文通过构造共享底层图和色多项式的非切换同构符号图回答了 Zaslavsky 的问题，推导了符号完全图连接的色多项式闭式表达，并引入双变量色多项式及其支配顶点删除递归公式，证明其在特定符号阈值图族中可刻画同构。
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…
We obtain a new weak Hilton-Milner type result for intersecting families of $k$-spaces in $\mathbb{F}_q^{2k}$, which improves several known results. In particular the chromatic number of the $q$-Kneser graph $qK_{n:k}$ was previously known for $n > 2k$ (except for $n=2k+1$ and $q=2$) or $k < q \log q - q$. Our result determines the chromatic number of $qK_{2k:k}$ for $q \geq 5$, so that the only remaining open cases are $(n, k) = (2k, k)$ with $q \in \{ 2, 3, 4 \}$ and $(n, k) = (2k+1, k)$ with $q = 2$.
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.
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by $8$ in the family of Sierpiński graphs with base $3$, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than $3$. On the other hand, we prove that the packing chromatic number in the family of …
On the Chromatic Number of Stable Kneser Hypergraphs: Verifying the Conjecture for New Families
本研究针对稳定 Kneser 超图色数猜想，在 s>r≥3 情形下，通过分析向量稳定 Kneser 图，首次对无穷多参数给出严格理论证明，覆盖 r=4, s=6 及 r=4, k=2, s=5 等新族。
We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specializes to (extended) $k$-deletion, and we give two methods to obtain numerous new bases from weighted graphs for the algebra of symmetric functions. Moreover, we classify when two weighted paths have equal extended chromatic symmetric functions by proving this is equivalent to the classification of equal ribbon Schur functions. This latter classification is dependent on the operation composition of compositions, which we generalize to composition of graphs. We then appl…
A Kneser representation KG(H) for a graph G is a bijective assignment of hyperedges of a hypergraph H to the vertices of G such that two vertices of G are adjacent if and only if the corresponding hyperedges are disjoint. In this paper, we introduce a colored version of the Turan number and use that to determine the chromatic number of some families of graphs in terms of the generalized Turan number of graphs. In particular, we determine the chromatic number of every Kneser multigraph KG(H), where the vertex set of H is the edge set of a multigraph G such that the multiplicity of each edge is …
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…
On Chromatic Asymptotic Approximate Groups
本文研究了近群的色渐近理论，结合Nathanson的色和集方法与近群理论中的渐近覆盖思想，提出了一种同时描述多个颜色类加法增长的框架，并在有限元组和线性集情况下建立了色覆盖定理，得出了更精确的二项界和整数设定下的渐近近群家族结构。
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind, respectively due to Simonyi and Tardos (Combinatorica, 2006), Simonyi, Tardif, and Zsbán (The Electronic Journal of Combinatorics, 2013), and Chen (Journal of Combinatorial Theory, Series A, 2011). As a consequence of the generalization of Chen's theorem, we get new families of graphs whose chromatic number equals their circular chromatic number and that satisf…
Tutte proved that if $G_{pt}$ is a planar triangulation and $P(G_{pt},q)$ is its chromatic polynomial, then $|P(G_{pt},τ+1)| \le (τ-1)^{n-5}$, where $τ=(1+\sqrt{5} \,)/2$ and $n$ is the number of vertices in $G_{pt}$. Here we study the ratio $r(G_{pt})=|P(G_{pt},τ+1)|/(τ-1)^{n-5}$ for a variety of planar triangulations. We construct infinite recursive families of planar triangulations $G_{pt,m}$ depending on a parameter $m$ linearly related to $n$ and show that if $P(G_{pt,m},q)$ only involves a single power of a polynomial, then $r(G_{pt,m})$ approaches zero exponentially fast as $n \to \inft…
We calculate the chromatic polynomials $P((G_s)_m,q)$ and, from these, the asymptotic limiting functions $W(\{G_s\},q)=\lim_{n \to \infty}P(G_s,q)^{1/n}$ for families of $n$-vertex graphs $(G_s)_m$ comprised of $m$ repeated subgraphs $H$ adjoined to an initial graph $I$. These calculations of $W(\{G_s\},q)$ for infinitely long strips of varying widths yield important insights into properties of $W(Λ,q)$ for two-dimensional lattices $Λ$. In turn, these results connect with statistical mechanics, since $W(Λ,q)$ is the ground state degeneracy of the $q$-state Potts model on the lattice $Λ$. For o…
The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $G$ be a graph of order $n$ and ${\rm rank}(G)$ be the rank of the adjacency matrix of $G$. In this paper we characterize all graphs with $E(G)={\rm rank}(G)$. Among other results we show that apart from a few families of graphs, $E(G)\geq 2\max(χ(G), n-χ(\bar{G}))$, where $n$ is the number of vertices of $G$, $\bar{G}$ and $χ(G)$ are the complement and the chromatic number of $G$, respectively. Moreover some new lower bounds for $E(G)$ in terms of ${\rm rank}(G)$ are given…
In 1995 Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$, whose $e$-positivity and Schur-positivity has been of large interest. In this paper we study the relative $e$-positivity and Schur-positivity between connected graphs on $n$ vertices. We define and investigate two families of posets on distinct chromatic symmetric functions. The relations depend on the $e$-positivity or Schur-positivity of a weighed subtraction between $X_G$ and $X_H$. We find a biconditional condition between $e$-positivity or Schur-positivity and the relation to the complete graph. This gives a…
$λ$-Chromatic Polynomials and Polytope Geometry
本文研究图的$λ$-色多项式，该多项式计数使用给定有限颜色集的不同$L(2,1)$-着色数量，通过内部-外部多面体框架下的格点枚举给出组合解释，并利用格路枚举和块-间隙技术计算完全图、完全二部图及多部图的$λ$-色多项式。
We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest $k$ for which there is an orientation, $D$, of $G$ with max indegree $k\!-\!1$ such that the number of even and odd circulations contained in $D$ are different. It is known that $χ(G) \leq χ_\ell(G) \leq χ_p(G) \leq AT(G)$, where $χ(G)$ is the chromatic number, $χ_\ell(G)$ is the list chromatic number, and $χ_p(G)$ is the paint number of $G$. In this paper we…
On the star b-chromatic number of a graph
本文引入图$G$的星b-色数$S_b(G)$，基于星着色集合上的严格偏序定义其为最小元素的最大颜色数，研究了其组合性质、已知图族的精确值及相关不变量比较。
Let $G$ be a graph. We introduce the acyclic b-chromatic number of $G$ as an analogue to the b-chromatic number of $G$. An acyclic coloring of a graph $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E(G)$ and the induced subgraph on vertices of any two colors $i,j\in \{1,\dots,k\}$ induces a forest. On the set of all acyclic colorings of $G$ we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number $A(G)$ of $G$ and the maximum cardinality of its minimal el…
We study a very large family of graphs, the members of which comprise disjoint paths of cliques with extremal cliques identified. This broad characterisation naturally generalises those of various smaller families of graphs having well-known chromatic polynomials. We derive a relatively simple formula for an arbitrary member of the subfamily consisting of those graphs whose constituent clique-paths have at least one trivial extremal clique, and use this formula to show that the set of all non-integer chromatic roots of these graphs is closed under multiplication by natural numbers. A well-know…
The chromatic symmetric function of a graph centred at a vertex
本文定义了以顶点为中心的色对称函数，利用NCSym和NCQSym商代数发现新的e-正单位区间图族，证实了Stanley-Stembridge猜想新情形，并细化了无环定向定理。
For a multigraph G, the integer round-up phi(G) of the fractional chromatic index yields a good general lower bound for the chromatic index . For an upper bound, Kahn showed that for any real c > 0 there exists a positive integer N so that the chromatic index is less than (1+c)*phi(G) whenever the fractional index > N. We show the amount by which the chromatic index can surpass phi(G) is in fact logarithmic, by showing that for any multigraph G with order n > 3 and at least one edge, the chromatic index is less than phi(G) + log (min {(n+1)/3, phi(G)}) .
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.
In this paper an improved bound on the chromatic number of the Pancake graph $P_n, n\geqslant 2$, is presented. The bound is obtained using a subadditivity property of the chromatic number of the Pancake graph. We also investigate an equitable coloring of $P_n$. An equitable $(n-1)$-coloring based on efficient dominating sets is given and optimal equitable $4$-colorings are considered for small $n$. It is conjectured that the chromatic number of $P_n$ coincides with its equitable chromatic number for any $n\geqslant 2$.
Upper bounds on the odd graceful chromatic number of graphs
本文研究二部图的奇优美色数，给出若干仅依赖顶点数或其二部划分相关图色数的新上界。
We provide new constructions of Meyniel extremal graphs, which are families of graphs with the conjectured largest asymptotic cop number. Using spanning subgraphs, we prove that there are an exponential number of new Meyniel extremal families with specified degrees. Using a linear programming problem on hypergraphs, we explore the degrees in families that are not Meyniel extremal. We give the best-known upper bound on the cop number of vertex-transitive graphs with a prescribed degree. We find new Meyniel extremal families of regular graphs with large chromatic number, large diameter, and expl…
Here we give refined numerical values for the minimum number of vertices of $k$-chromatic unit distance graphs in the Euclidean plane.
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pend…
Chromatic numbers with open and nonzero local modular constraints
该文研究带局部模约束的图色数问题，给出了闭/开邻域和模n非零约束下对应色数的界，以及开邻域和模n同余k时最小着色阶数的界与部分图族的精确值。
The odd chromatic number and the conflict-free chromatic number are new graph parameters introduced by Petruševski and Škrekovski [2021] and Fabrici, Lužar, Rindošová and Soták [2022] respectively. In this note, we show that graphs with bounded $2$-strong colouring number have bounded odd chromatic number and bounded conflict-free chromatic number. This implies that graph classes with bounded expansion have bounded odd chromatic number and bounded conflict-free chromatic number. Moreover, it follows by known results that the odd chromatic number and the conflict-free chromatic number of $k$-pl…
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.
Let $G$ be a circle graph without clique on 4 vertices. We prove that the chromatic number of $G$ doesn't exceed 30.
We show that for a simple graph $G$, $c'(G)\leqΔ(G)+2$ where $c'(G)$ is the choice index (or edge-list chromatic number) of $G$, and $Δ(G)$ is the maximum degree of $G$. As a simple corollary of this result, we show that the total chromatic number $χ_T(G)$ of a simple graph satisfies the inequality $χ_T(G)\leq\ Δ(G)+4$ and the total choice number $c_T(G)$ also satisfies this inequality. We also relate these bounds to the Hall index and the Hall condition index of a simple graph, and to the total Hall number and the total Hall condition number of a simple graph.
Let $G$ be a graph with maximum degree $Δ$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours admitting such $c$ is the conflict-free chromatic index of $G$, denoted by $χ'_{CF}(G)$. In "Conflict-free chromatic number versus conflict-free chromatic index" [J. Graph Theory, 2022; 99: 349--358] it was recently proved by means of the probabilistic method that $χ'_{CF}(G)\leq C_1\log_2Δ+C_2$, where $C_1>337$ and $C_2$ are constants, whereas there are families of gra…
The anagram-free chromatic number is a new graph parameter introduced independently Kamčev, Łuczak, and Sudakov (2017) and Wilson and Wood (2017). In this note, we show that there are planar graphs of pathwidth 3 with arbitrarily large anagram-free chromatic number. More specifically, we describe $2n$-vertex planar graphs of pathwidth 3 with anagram-free chromatic number $Ω(\log n)$. We also describe $kn$ vertex graphs with pathwidth $2k-1$ having anagram-free chromatic number in $Ω(k\log n)$.
The $k^{\text{th}}$ power of a graph $G=(V,E)$, $G^k$, is the graph whose vertex set is $V$ and in which two distinct vertices are adjacent if and only if their distance in $G$ is at most $k$. This article proves various eigenvalue bounds for the independence number and chromatic number of $G^k$ which purely depend on the spectrum of $G$, together with a method to optimize them. Our bounds for the $k$-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the…
Let $χ(G$) and $χ_f(G)$ denote the chromatic and fractional chromatic numbers of a graph $G$, and let $(n^+ , n^0 , n^-)$ denote the inertia of $G$. We prove that: \[ 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le χ(G) \mbox{ and conjecture that } 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le χ_f(G) \] We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with a discussion of asymmetry between $n^+$ and $n^-$, including some Nordhaus-Gaddum bounds for inertia.
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…
Infinite families of planar graphs of a given injective chromatic number
本研究通过计算验证与无限族3-连通平面图构造，拓展了平面图单射着色相关猜想的研究成果，给出多个经典猜想的无限反例并为修正猜想提供支撑。
An ordered graph $G$ is a graph together with a specified linear ordering on the vertices, and its interval chromatic number is the minimum number of independent sets consisting of consecutive vertices that are needed to partition the vertex set. The $t$-color Ramsey number $R_t(G)$ of an ordered graph $G$ is the minimum number of vertices of an ordered complete graph such that every edge-coloring from a set of $t$ colors contains a monochromatic copy of $G$ such that the copy of $G$ preserves the original ordering on $G$. An ordered graph is $k$-ichromatic if it has interval chromatic number …
Circular chromatic index of small graphs
本研究系统确定了最大度数为4、5、6的小图和多重图的环色数，并推翻了关于环色数接近Δ+1的猜想。
A logarithmic bound for the chromatic number of the associahedron
本文研究结合多面体色数增长问题，证明其色数关于维数具有对数上界，改进旧结果并证实2009年猜想。
In Stanley's seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not $e$-positive, namely, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not $e$-positive. Moreover, one such family is additionally claw-free, thus establishing that the $e$-positivity of chromatic symmetric functions is in general not dependent on th…
Here we prove that a graph without some three induced subgraphs has chromatic number at the most equal to its maximum clique size plus one. Further we show that the bounds are tight and give examples to show that each of the three forbidden subgraphs is necessary in the hypothesis.
The Neighbor-Locating-Chromatic Number of Pseudotrees
本文研究伪树的邻点定位色数，确定路径、环、扇形图和轮形图的该参数并给出最小着色构造，同时得到树与单圈图阶数的紧上界。
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