Dongseok Kim, Jaeun Lee
If we fix a spanning subgraph $H$ of a graph $G$, we can define a chromatic number of $H$ with respect to $G$ and we show that it coincides with the chromatic number of a double covering of $G$ with co-support $H$. We also find a few estimations for the chromatic numbers of $H$ with respect to $G$.
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Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \medskip \noindent $\bullet$ If the bicliques partition the edges of $G$, then their number is at least $2^{\sqrt{\log_2 k}}$. This is the first improvement of the easy lower bound of $\log_2 k$, while the Alon-Saks-Seymour conjecture states that this can be improved to $k-1$. \medskip \noindent $\bullet$ The sum of the orders of the bicliques is at least $(1-o(1))k\log_2 k$. This generalizes, in asymptotic form, a result o…
A connected $k$-chromatic graph $G$ is said to be {\it double-critical} if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. A longstanding conjecture of Erdős and Lovász states that the complete graphs are the only double-critical graphs. Kawarabayashi, Pedersen and Toft [\it{Electron. J. Combin.}, 17(1): Research Paper 87, 2010] proved that every double-critical $k$-chromatic graph with $k \leq 7$ contains a $K_k$ minor. It remains unknown whether an arbitrary double-critical $8$-chromatic graph contains a $K_8$ minor, but in this paper we prove that any double-critical …
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。
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.
A connected $n$-chromatic graph $G$ is double-critical if for all the edges $xy$ of $G$, the graph $G-x-y$ is $(n-2)$-chromatic. In 1966, Erd\H os and Lovász conjectured that the only double-critical $n$-chromatic graph is $K_n$. This conjecture remains unresolved for $n \ge 6.$ In this short note, we verify this conjecture for claw-free graphs $G$ of chromatic number $6$.
A double-logarithmic upper bound on the chromatic number of the associahedron
本文证明了结合多面体的色数上界从 O(log n) 改进到 O(log log n)，主要贡献是给出了更紧的色数上界。
A connected $t$-chromatic graph $G$ is \dfn{double-critical} if $G \backslash\{u, v\}$ is $(t-2)$-colorable for each edge $uv\in E(G)$. A long standing conjecture of Erdős and Lovász that the complete graphs are the only double-critical $t$-chromatic graphs remains open for all $t\ge6$. Given the difficulty in settling Erdős and Lovász's conjecture and motivated by the well-known Hadwiger's conjecture, Kawarabayashi, Pedersen and Toft proposed a weaker conjecture that every double-critical $t$-chromatic graph contains a $K_t$ minor and verified their conjecture for $t\le7$. Albar and Gonçalves…
We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that circular cliques play for the circular chromatic number. The fact that the defined parameter attains on these graphs the `correct' value suggests that the definition is a n…
Assume $n, m$ are positive integers and $G$ is a graph. Let $P_{n,m}$ be the graph obtained from the path with vertices $\{-m, -(m-1), \ldots, 0, \ldots, n\}$ by adding a loop at vertex $ 0$. The double cone $Δ_{n,m}(G)$ over a graph $G$ is obtained from the direct product $G \times P_{n,m}$ by identifying $V(G) \times \{n\}$ into a single vertex $(\star, n)$, identifying $V(G) \times \{-m\}$ into a single vertex $(\star, -m)$, and adding an edge connecting $(\star, -m)$ and $(\star, n)$. This paper determines the fractional chromatic number of $Δ_{n,m}(G)$. In particular, if $n < m$ or $n=m$ …
The acyclic chromatic index of a graph $G$ is the least number of colors needed to properly color its edges so that none of its cycles is bichromatic. In this work, we show that $2Δ-1$ colors are sufficient to produce such a coloring, where $Δ$ is the maximum degree of the graph. In contrast with most extant randomized algorithmic approaches to the chromatic index, where the algorithms presuppose enough colors to guarantee properness deterministically and use randomness only to deal with the bichromatic cycles, our randomized, Moser-type algorithm produces a not necessarily proper random color…
A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. The only known double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The conjecture that there are no other double-critical graphs is a special case of a conjecture from 1966, due to Erdős and Lovász. The conjecture has been verified for $k \leq 5$. We prove for $k=6$ and $k=7$ that any non-complete double-critical $k$-chromatic graph is 6-connected and has $K_k$ as a minor.
A connected graph $G$ with chromatic number $t$ is double-critical if $G \backslash \{x, y\}$ is $(t - 2)$-colorable for each edge $xy \in E(G)$. The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erd\H os and Lovász from 1966, which is referred to as the Double-Critical Graph Conjecture, states that there are no other double-critical graphs. That is, if a graph $G$ with chromatic number $t$ is double-critical, then $G$ is the complete graph on $t$ vertices. This has been verified for $t \le 5$, but remains open for $t \ge 6$. In this paper…
The Chromatic Number of the Disjointness Graph of the Double Chain
研究双链点配置中不相交线段着色的最小颜色数，证明所需颜色数为上下链点数之和减去一个基于下链点数的下取整表达式，且该数充分。
An $n$-lift of a graph $G$ is a graph from which there is an $n$-to-$1$ covering map onto $G$. Amit, Linial, and Matou\v sek (2002) raised the question of whether the chromatic number of a random $n$-lift of $K_5$ is concentrated on a single value. We consider this problem for $G=K_{d+1}$, and show that for fixed $d\ge 3$ the chromatic number of a random lift of $K_d$ is (asymptotically almost surely) either $k$ or $k+1$, where $k$ is the smallest integer satisfying $d < 2k \log k$. Moreover, we show that, for roughly half of the values of $d$, the chromatic number is concentrated on $k$. The …
The bounds on the sum and product of chromatic numbers of a graph and its complement are known as Nordhaus-Gaddum inequalities. In this paper, we study the operations on the Independence numbers of graphs with their complement. We also provide a new characterization of certain graph classes.
Hofmman's bound on the chromatic number of a graph states that $χ\geq 1 - \frac {λ_1} {λ_n}$. Here we show that the same bound, or slight modifications of it, hold for several graph parameters related to the chromatic number: the vector coloring number, the $ψ$-covering number and the $λ$-clustering number.
We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for $t\leq \frac38p+1$, a small …
We investigate the $e$-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods involve extracting particular $e$-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott's triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We also propose some conjectures on the $e$-positivity and Schur positivity of trees.
An $r$-segment hypergraph $H$ is a hypergraph whose edges consist of $r$ consecutive integer points on line segments in $\mathbb{R}^2$. In this paper, we bound the chromatic number $χ(H)$ and covering number $τ(H)$ of hypergraphs in this family, uncovering several interesting geometric properties in the process. We conjecture that for $r \ge 3$, the covering number $τ(H)$ is at most $(r - 1)ν(H)$, where $ν(H)$ denotes the matching number of $H$. We prove our conjecture in the case where $ν(H) = 1$, and provide improved (in fact, optimal) bounds on $τ(H)$ for $r \le 5$. We also provide sharp bo…
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 prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that the edge-chromatic number is the maximum degree, except when simple arithmetic forces the edge-chromatic number to be one greater than the maximum degree. For Mycielski graphs, we strengthen an old result that the graphs are Hamiltonian by showing t…
Eight times four bialgebras of hypergraphs, cointeractions, and chromatic polynomials
该研究推广Schmitt图Hopf代数得到超图双代数，证明其为Foissy意义下的双代数且对应超图色多项式，结合超图的对偶、补等对合衍生结构，共给出八组四类双代数，包含多位学者近期提出的相关结构。
The chromatic number of a digraph $D$ is the minimum number of acyclic subgraphs covering the vertex set of $D$. A tournament $H$ is a hero if every $H$-free tournament $T$ has chromatic number bounded by a function of $H$. Inspired by the celebrated Erdős--Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. We extend this framework to dense digraphs: A digraph $H$ is a superhero if every $H$-free digraph $D$ has chromatic number bounded by a function of $H$ and $α(D)$, the independence number of the underlying graph of $D$. We prove here that a digraph is a super…
Given a graph $G$, its orientation covering number $σ(G)$ is the smallest non-negative integer $k$ with the property that we can choose $k$ orientations of $G$ such that whenever $x, y, z$ are vertices of $G$ with $xy,xz\in E(G)$ then there is a chosen orientation in which both $xy$ and $xz$ are oriented away from $x$. Esperet, Gimbel and King showed that $σ(G)\leq σ\left(K_{χ(G)}\right)$, where $χ(G)$ is the chromatic number of $G$, and asked whether we always have equality. In this note we prove that it is indeed always the case that $σ(G)=σ(K_{χ(G)})$. We also determine the exact value of $…
Spectral bounds for distance coloring and packing parameters of graphs via semidefinite programming
利用谱图论和半定规划方法，获得了与距离染色和打包相关的几个图参数的尖锐谱界，包括内射色数、开打包数、内射色指数和强色指数，并应用于超立方体的码覆盖数和开打包数，得到新的精确值和加强界。
Coloring, list coloring, and fractional coloring in intersections of matroids
研究k个拟阵交集超图的着色参数差距，证明列表色数不超过色数的k倍及最大色数的2k-1倍，并借助拓扑工具建立多面体与分数色数、匹配及覆盖数的联系。
Let $G$ be a finite $d$-regular graph with a proper edge coloring. An edge Kempe switch is a new proper edge coloring of $G$ obtained by switching the two colors along some bi-chromatic cycle. We prove that any other edge coloring can be obtained by performing finitely many edge Kempe switches, provided that $G$ is replaced with a suitable finite covering graph. The required covering degree is bounded above by a constant depending only on $d$.
The proper divisor graph $Υ_n$ of a positive integer $n$ is the simple graph whose vertices are the proper divisors of $n$, and in which two distinct vertices $u, v$ are adjacent if and only if $n$ divides $uv$. The graph $Υ_n$ plays an important role in the study of the zero divisor graph of the ring $\mathbb{Z}_n$. In this paper, we study some graph theoretic properties of $Υ_n$ and determine the graph parameters such as clique number, chromatic number, chromatic index, independence number, matching number, domination number, vertex and edge covering numbers of $Υ_n$. We also determine the a…
On the chromatic numbers of Johnson type graphs
本文研究Johnson型图的色数增长阶，确定了 $J_\pm(n,2,-1)$ 和 $J_\pm(n,3,-1)$ 为对数阶增长，$J_\pm(n,3,-2)$ 为双重对数阶增长。
Graph coverings are known to induce surjections of their critical groups. Here we describe the kernels of these morphisms in terms of data parametrizing the covering. Regular coverings are parametrized by voltage graphs, and the above kernel can be identified with a naturally defined voltage graph critical group. For double covers, the voltage graph is a signed graph, and the theory takes a particularly pleasant form, leading also to a theory of double covers of signed graphs.
The neighborhood complex $N(G)$ is a simplicial complex assigned to a graph $G$ whose connectivity gives a lower bound for the chromatic number of $G$. We show that if the Kronecker double coverings of graphs are isomorphic, then their neighborhood complexes are isomorphic. As an application, for integers $m$ and $n$ greater than 2, we construct connected graphs $G$ and $H$ such that $N(G) \cong N(H)$ but $χ(G) = m$ and $χ(H) = n$. We also construct a graph $KG_{n,k}'$ such that $KG_{n,k}'$ and the Kneser graph $KG_{n,k}$ are not isomorphic but their Kronecker double coverings are isomorphic.
The main eigenvalues of a graph $G$ are those eigenvalues of the $(0,1)$-adjacency matrix $\mathbf A$ having a corresponding eigenvector not orthogonal to $\mathbf j = (1,\dots,1)$. The CDC of a graph $G$ is the direct product $G\times K_2$. The main eigenspace of $\mathbf A$ is generated by the principal main eigenvectors and is the same as the image of the walk matrix. A hierarchy of properties of pairs of graphs is established in view of their CDC's, walk matrices, main eigenvalues, eigenvectors and eigenspaces. We determine by algorithm that there are 32 pairs of non-isomorphic graphs on a…
We study the generalized Turán function $ex(n,H,F)$, when $H$ or $F$ is a double star $S_{a,b}$, which is a tree with a central edge $uv$, $a$ leaves connected to $u$ and $b$ leaves connected to $v$. We determine $ex(n,K_k,S_{a,b})$ and $ex(n,S_{a,b},F)$ for sufficiently large $n$, where $F$ is either a 3-chromatic graph with an edge whose deletion results in a bipartite graph, or the 2-fan, i.e. two triangles sharing a vertex. We also give bounds on $ex(n,S_{a,b},S_{c,d})$.
In 1995, Erdős and Gyárfás proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2\sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers $r,s$, what is the smallest number $\text{pc}_{r,s}(K_n)$ such that in every colouring of the edges of $K_n$ with $r$ colours, there exists a vertex cover of $K_n$ by $\text{pc}_{r,s}(K_n)$ monochromatic paths using altogether at most $s$ different colours? For fixed i…
An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to prove a conjecture of Graber and Pandharipande, giving a linear recurrence equation for the number of these coverings with no ramification over infinity. The general form of the series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.
The Fortuin-Kasteleyn polynomial as a bialgebra morphism and applications to the Tutte polynomial
本文证明 Fortuin–Kasteleyn 多项式是图双代数上的态射，并借助全无环部分定向、覆盖结构及所有定向映射，给出其反极元公式与 Tutte 多项式的组合解释。
In this paper, following the methods of http://arxiv.org/abs/1305.7449v2, we show that the double covering groups of the symmetric and alternating groups have p-basic sets for any odd prime p.
We give a bijective proof of Hurwitz formula for the number of simple branched coverings of the sphere by itself. Our approach extends to double Hurwitz numbers and yields new properties for them. In particular we prove for double Hurwitz numbers a conjecture of Kazarian and Zvonkine, and we give an expression that in a sense interpolates between two celebrated polynomiality properties: polynomiality in chambers for double Hurwitz numbers, and a new analog for almost simple genus 0 Hurwitz numbers of the polynomiality up to normalization of simple Hurwitz numbers of genus $g$. Some probabilist…
The game chromatic number of generalized Mycielski graphs of paths and cycles
针对路径和圈的广义Mycielski图，确定其游戏色数介于4至5之间，得出M_2(P_5)与M_2(P_6)的游戏色数精确值为4，证明其不随层数和基础图阶数增长而超过5。
On asymptotic packing of convex geometric and ordered graphs
本文研究凸几何图与有序图的渐近填充问题，证明循环色数至多4的凸几何图及区间色数至多3的有序图均可填充，并说明该界最优。
Boundedness and Separation in the Graph Covering Number Framework
本文研究图覆盖数框架中四个覆盖数之间的有界性与分离性，针对不同图类组合，给出绑定函数或证明其不存在。
Covering Families for DP-Coloring of Cartesian Products with Complete Bipartite Graphs
本文引入覆盖族概念，证明其最小大小与DP-染色中完全二部图笛卡尔积的临界参数相等，从而证明μ(4)=12并改进上下界。
Local Clique Covers and Chromatic Number
本文证明了局部团覆盖数与色数之和不超过顶点数加一的猜想，并给出了达到等号时图的结构性质及诱导匹配的改进界。
Coloring Geometric Hypergraphs: A Survey
本文综述了几何超图的着色问题，探讨了色数与覆盖问题的关系，以及当m足够大时色数是否为2的结论。
Difference Methods for Double-Change Covering Designs
本文研究双变覆盖设计（DCCD），提出基于1-因子分解和扩展集的递归构造方法，并利用差分方法构造了五个极小循环DCCD的无限族及多个具体极小设计。
Boundedness and Separation Between Induced and Non-Induced Covering Numbers
本文研究诱导与非诱导覆盖数之间的有界性，针对不同性质的图类，在240种组合中为219种确定了绑定函数的存在性，并证明遗传客类在三个覆盖数上总有绑定函数。
Distinguishing colorings, proper colorings, and covering properties without the Axiom of Choice
本文在无选择公理的ZF集合论中证明若干关于无限局部有限连通图的色数、边色数、区分数与区分指数的命题等价于Kőnig引理，并给出与AC或Kőnig引理等价的覆盖与支配集存在性新条件。
Hypergraph Universality and Erdos-Posa Failure for Chromatically Rich Odd Cycles
本文证明对任意k≥3，任意有限丛可由k-坏奇圈的极小终端迹实现，保持打包-覆盖数，导致有限拥塞和分数Erdős-Pósa性质失效，与k=2情形形成对比。
We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.
In this paper we prove an asymptotic multipartite version of a well-known theorem of Kühn and Osthus by establishing, for any graph $H$ with chromatic number $r$, the asymptotic multipartite minimum degree threshold which ensures that a large $r$-partite graph $G$ admits a perfect $H$-tiling. We also give the threshold for an $H$-tiling covering all but a linear number of vertices of $G$, in a multipartite analogue of results of Komlós and of Shokoufandeh and Zhao.
Betti numbers and torsions in homology groups of double coverings
本文针对双重覆盖空间，证明排列的 Papadima–Suciu 严格不等式与其 Milnor 纤维第一整同调群存在非平凡 $2$-挠率彼此等价。
A recent paper of Balogh, Li and Treglown initiated the study of Dirac-type problems for ordered graphs. In this paper we prove a number of results in this area. In particular, we determine asymptotically the minimum degree threshold for forcing (i) a perfect $H$-tiling in an ordered graph, for any fixed ordered graph $H$ of interval chromatic number at least $3$; (ii) an $H$-tiling in an ordered graph $G$ covering a fixed proportion of the vertices of $G$ (for any fixed ordered graph $H$); (iii) an $H$-cover in an ordered graph (for any fixed ordered graph $H$). The first two of these results…
Let $ H = (V,E) $ be a hypergraph. By the chromatic number of a hypergraph $ H = (V,E) $ we mean the minimum number $χ(H)$ of colors needed to paint all the vertices in $ V $ so that any edge $ e \in E $ contains at least two vertices of some different colors. Finally, a hypergraph is said to form a clique, if its edges are pairwise intersecting. In 1973 Erdős and Lovász noticed that if an $n$-uniform hypergraph $ H = (V,E) $ forms a clique, then $ χ(H) \in \{2,3\} $. They untoduced following quantity. $$ M(n) = \max \{|E|: \exists {\rm an} n-{\rm uniform} {\rm clique} H = (V,E) {\rm with} χ(H…
Covering Bridgeless Claw-Free Cubic Graphs with Perfect Matchings
本文证明Berge猜想对无桥无爪三次图成立，并给出四个完美匹配覆盖至少299/315边数的结果。
The local antimagic (total) chromatic numbers of firecracker graphs and edge-corona product graphs
本文研究了火柴人图和边冠状积图的局部反魔法色数，通过拼接星图和构造边冠状积图，确定了其色数的计算方法。
The double Hall property and cycle covers in bipartite graphs
该研究围绕二分图双Hall性质展开，证明满足该性质的二分图存在覆盖A的2-因子，对B中顶点度数受限的图验证了Salia猜想，还给出边数的紧下界。
Hopf-algebraic structures on mixed graphs
本文在混合图上引入两种余积构建双余代数结构，证明其唯一多项式不变量为强色多项式，并揭示了其与弱色多项式等的联系及无环定向的代数证明。
Strong Colouring of the Qualitative Independence Hypergraph $3\text{-}QI(11,2)$
研究定性独立超图$3\text{-}QI(11,2)$的强着色，通过将顶点视为特定子集并评估相交集系统确定强独立数，利用相关界确定覆盖阵列数为11并给出超图满足此条件的充分条件。
Exponentially Many Circuit Double Covers
研究立方图回路双覆盖的计数版本，通过线性方程组系统，证明2边连通3边可着色立方图至少有2^{n/2 - 1}个回路双覆盖，给出围长至少16的3边连通立方图的较弱指数下界，还刻画了立方图平面性及给出5回路双覆盖的流条件。
On generalized covering radii of binary primitive double-error-correcting BCH codes
本文通过广义超码引理简化了二进制原始双错误纠正BCH码的广义覆盖半径下界证明，并建立了新的下界，同时结合组合论与Weil型指数和估计，证明了广义覆盖半径的范围。
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