Maria Chudnovsky, Alex Scott, Paul Seymour
We prove a 1985 conjecture of Gyárfás that for all $k,\ell$, every graph with sufficiently large chromatic number contains either a complete subgraph with $k$ vertices or an induced cycle of length at least $\ell$.
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Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.
It is well known that a random subgraph of the complete graph $K_n$ has chromatic number $Θ(n/\log n)$ w.h.p. Boris Bukh asked whether the same holds for a random subgraph of any $n$-chromatic graph, at least in expectation. In this paper it is shown that for every graph, whose fractional chromatic number is at least $n$, the fractional chromatic number of its random subgraph is at least $n/(8\log_2(4n))$ with probability more than $1-\frac{1}{2n}$. This gives the affirmative answer for a strengthening of Bukh's question for the fractional chromatic number.
A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for every positive integer k, every triangle-free graph with sufficiently large chromatic number contains holes of k consecutive lengths.
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.
We prove a conjecture of Andras Gyarfas, that for all k,t, every graph with clique number at most k and sufficiently large chromatic number has an odd hole of length at least t.
An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyarfas made the conjecture that for all t there exists n such that every graph with no K_t subgraph and no odd hole is n-colourable. We prove this conjecture.
Although the chromatic number of a graph is not known in general, attempts have been made to find good bounds for the number. Here we prove that for a graph G with two forbidden subgraphs and maximum degree less than or equal to 2ω - 3, χ equals its maximum clique size. We also give examples to show that the condition is necessary.
A large body of research in graph theory concerns the induced subgraphs of graphs with large chromatic number, and especially which induced cycles must occur. In this paper, we unify and substantially extend results from a number of previous papers, showing that, for every positive integer k, every graph with large chromatic number contains either a large complete subgraph or induced cycles of all lengths modulo k. As an application, we prove two conjectures of Kalai and Meshulam from the 1990's connecting the chromatic number of a graph with the homology of its independence complex.
Investigating the equality of the chromatic number and the circular chromatic number of graphs has been an active stream of research for last decades. In this regard, Habolhassan and Zhu [Circular chromatic number of Kneser graphs, Journal of Combinatorial Theory Series B, 2003] proved that if $n$ is sufficiently large with respect to $k$, then the Schrijver graph ${\rm SG}(n,k)$ has the same chromatic and circular chromatic number. Later, Meunier [A topological lower bound for the circular chromatic number of Schrijver graphs, Journal of Graph Theory, 2005] and independently, Simonyi and Tard…
On the solution to the Erdős-Hajnal problem on high-girth high-chromatic subgraphs
本文解释并优化了 Erdős-Hajnal 问题的高围长高色数子图的否定构造，将 Kohlmeyer 和 Kruer 的色数上界从 6 改进到 3。
We prove that for all nonnegative integers k,s there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. This extends a recent result of Gyarfas and Sarkozy, who proved the same (when k=2) for graphs G with girth at least five.
The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are $2$-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an $H$-avoiding $k$-coloring to be a proper $k$-coloring that forbid…
For an oriented graph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$. Let $f(n)$ be the smallest integer such that every oriented graph $G$ with chromatic number larger than $f(n)$ has $f(G) > n$. Let $g(n)$ be the smallest integer such that every tournament $G$ with more than $g(n)$ vertices has $f(G) > n$. It is straightforward that $Ω(n) \le g(n) \le f(n) \le n^2$. This paper provides the first nontrivial lower and upper bounds for $g(n)$. In particular, it is proved that $\frac{1}{4}n^{8/7} \le g(n) \le n^2-(2-\frac{1}{\sqrt{2}})n+2$. It is also shown tha…
Resolving a problem raised by Norin, we show that for each $k \in \mathbb{N}$, there exists an $f(k) \le 7k$ such that every graph $G$ with chromatic number at least $f(k)+1$ contains a subgraph $H$ with both connectivity and chromatic number at least $k$. This result is best-possible up to multiplicative constants, and sharpens earlier results of Alon-Kleitman-Thomassen-Saks-Seymour from 1987 showing that $f(k) = O(k^3)$, and of Chudnovsky-Penev-Scott-Trotignon from 2013 showing that $f(k) = O(k^2)$. Our methods are robust enough to handle list colouring as well: we also show that for each $k…
Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1) Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; (2) For all $l$, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than $l$; (3) For all $k$, $l$, every graph with no clique of size more than $k$ and sufficiently large chroma…
Triangle-free subgraphs with large fractional chromatic number
本文证明了 Erdős-Hajnal 猜想中围长为4情形的分数色数版本，即存在函数 h(k,4)，使色数足够大的图必含分数色数至少为 k 的无三角形子图。
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 …
Induced subgraphs of induced subgraphs of large chromatic number
本文证明对于任意有边的图$F$，存在色数任意大且团数相同的图，其$F$-free导出子图色数有界，推广了多项近期定理及Folkman结果，并扩展至竞赛图、超图等结构。
Fix an oriented graph H, and let G be a graph with bounded clique number and very large chromatic number. If we somehow orient its edges, must there be an induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for two specific kinds of digraph H: the three-edge path, with the first and last edges both directed towards the interior; and stars (with many edges directed out and many directed in). Aboulker et al subsequently conjectured that the answer is affirmative in both cases. We give affirmative answers to both questions.
Acyclic subgraphs of tournaments with high chromatic number
本文证明了每个 $n$ 顶点竞赛图存在色数至少为 $n^{5/9-o(1)}$ 的无环子图，且上界为 $n^{3/4+o(1)}$，强有力地证实并改进了 Nassar 和 Yuster 的猜想，其证明结合了概率与谱技术。
We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper, one of us proved that every tree has this property; and in another earlier paper with M. Chudnovsky, we proved that every cycle has this property. Here we give a common generalization. Say a banana is the union of a set of paths all with the same ends but otherwise disjoint. We prove that if H is obtained from a tree by replacing each edge by a banana then H has the property mentio…
We prove that for every $n$, there is a graph $G$ with $χ(G) \geq n$ and $ω(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $ω(H) \leq 2$ satisfies $χ(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.
Chromatic number and regular subgraphs
本文否定了Erdős和Hajnal关于高色数图必含同顶点集上多条边不相交环的猜想，构造了分数色数较高但不含4-正则子图的图，并证明该界在Harris猜想下最优。
We prove that, for every function $f:\mathbb{N} \rightarrow \mathbb{N}$, there is a graph $G$ with uncountable chromatic number such that, for every $k \in \mathbb{N}$ with $k \geq 3$, every subgraph of $G$ with fewer than $f(k)$ vertices has chromatic number less than $k$. This answers a question of Erdős, Hajnal, and Szemeredi.
We give an elementary proof for the non-$3$-colorability of $4$-chromatic Schrijver graphs thus providing such a proof also for $4$-chromatic Kneser graphs. To this end we use a complete description of the structure of $4$-chromatic Schrijver graphs that was already given by Braun and even earlier in an unpublished manuscript by Li. We also address connections to surface quadrangulations. In particular, we show that a spanning subgraph of $4$-chromatic Schrijver graphs quadrangulates the Klein bottle, while another spanning subgraph quadrangulates the projective plane. The latter is a special …
A class of graphs is $χ$-bounded if there is a function $f$ such that $χ(G)\le f(ω(G))$ for every induced subgraph $G$ of every graph in the class, where $χ,ω$ denote the chromatic number and clique number of $G$ respectively. In 1987, Gyárfás conjectured that for every $c$, if $\mathcal{C}$ is a class of graphs such that $χ(G)\le ω(G)+c$ for every induced subgraph $G$ of every graph in the class, then the class of complements of members of $\mathcal{C}$ is $χ$-bounded. We prove this conjecture. Indeed, more generally, a class of graphs is $χ$-bounded if it has the property that no graph in th…
Induced Subgraph Bounds on the Zero Forcing Number and Chromatic Consequences
研究关于图的色数、团数和零强制数的关系，通过无三角形诱导子图建立零强制数下界，提出\((χ, ω, Z)\)猜想，证明其对无三角形正则图成立并提供数值证据。
Maximum odd induced subgraph of a graph concerning its chromatic number
本文研究图的奇诱导子图最大阶数问题，证明 Scott 猜想对二部图不成立但对线图成立，并否定 Berman 等人的相关猜想。
Gyárfás and Sumner independently conjectured that for every tree $T$, the class of graphs not containing $T$ as an induced subgraph is $χ$-bounded, that is, the chromatic numbers of graphs in this class are bounded above by a function of their clique numbers. This remains open for general trees $T$, but has been proved for some particular trees. For $k\ge 1$, let us say a broom of length $k$ is a tree obtained from a $k$-edge path with ends $a,b$ by adding some number of leaves adjacent to $b$, and we call $a$ its handle. A tree obtained from brooms of lengths $k_1,...,k_n$ by identifying thei…
Quantum Chromatic Number of Subgraphs of Orthogonality Graphs and the Distance-2 Hamming Graph
本文研究了正交图子图和距离-2汉明图的量子色数，利用组合设计技术确定了它们的精确值。
Two subgraphs $A,B$ of a graph $G$ are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if $G$ is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there should be two anticomplete subgraphs both with large minimum degree, and that is one of our results. We prove two variants of this. First, a strengthening: we can ask for one of the two subgraphs to have …
We consider the structure of $H$-free subgraphs of graphs with high minimal degree. We prove that for every $k>m$ there exists an $ε:=ε(k,m)>0$ so that the following holds. For every graph $H$ with chromatic number $k$ from which one can delete an edge and reduce the chromatic number, and for every graph $G$ on $n>n_0(H)$ vertices in which all degrees are at least $(1-ε)n$, any subgraph of $G$ which is $H$-free and contains the maximum number of copies of the complete graph $K_m$ is $(k-1)$-colorable. We also consider several extensions for the case of a general forbidden graph $H$ of a given …
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.
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.
It is known that every graph of sufficiently large chromatic number and bounded clique number contains, as an induced subgraph, a subdivision of any fixed forest, and a subdivision of any fixed cycle. Equivalently, forests and triangles are pervasive, where H is pervasive (in some class of graphs) if for all s>0, every graph in the class with bounded clique number and sufficiently large chromatic number contains an induced subdivision of H, with every edge subdivided at least s times. Which other graphs are pervasive? Chalopin, Esperet, Li and Ossona de Mendez proved that every such graph is a…
Let $n,k,r$ be positive integers with $n \geq rk$ and $r \geq 2$. Consider a circle $C$ with~$n$ points~$1,\ldots,n$ in clockwise order. The $r$-stable \emph{interlacing graph} $\text{IG}_{n,k}^{(r)}$ is the graph with vertices corresponding to $k$-subsets $S$ of $\{1,...,n\}$ such that any two distinct points in~$S$ have distance at least~$r$ around the circle, and edges between~$k$-subsets $P$ and $Q$ if they \emph{interlace}: after removing the points in~$P$ from $C$, the points in~$Q$ are in different connected components. In this paper we prove that the circular chromatic number of $\text…
Given a graph $G$ and $p \in [0,1]$, let $G_p$ denote the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. Alon, Krivelevich, and Sudokov proved $\mathbb{E} [χ(G_p)] \geq C_p \frac{χ(G)}{\log |V(G)|}$, and Bukh conjectured an improvement of $\mathbb{E}[χ(G_p)] \geq C_p \frac{χ(G)}{\log χ(G)}$. We prove a new spectral lower bound on $\mathbb{E}[χ(G_p)]$, as progress towards Bukh's conjecture. We also propose the stronger conjecture that for any fixed $p \leq 1/2$, among all graphs of fixed chromatic number, $\mathbb{E}[χ(G_p)]$ is minimized by the complet…
A note on vertex-critical induced subgraphs of shift graphs
本文研究了移位图中顶点临界诱导子图的性质，发现每个k色移位图包含唯一的k-顶点临界子图，并给出了其显式描述。
We discuss some problems related to induced subgraphs. The first problem is about getting a good upper bound for the chromatic number in terms of the clique number for graphs in which every induced cycle has length $3$ or $4$. The second problem is about the perfect chromatic number of a graph, which is the smallest number of perfect sets into which the vertex set of a graph can be partitioned. (A set of vertices is said to be perfect it it induces a perfect graph.) The third problem is on antichains in the induced subgraph ordering. The fourth problem is on graphs in which the difference betw…
The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic number is upper bounded by a function of the Hall ratio. We answer this question in negative, by showing two results of independent interest regarding 1-subdivisions (the 1-subdivision of a graph is obtained by subdividing each edge exactly once). * For every c > 0, every graph of sufficiently large average degree contains as a subgraph the 1-subdivision of…
Counting subgraphs of coloring graphs
本文将色多项式推广为计数着色图中任意导出子图的$π_G^{(H)}(k)$，证明其关于颜色数$k$为多项式，研究了色对多项式并揭示了其与树度序列的等价关系，同时生成了新的图不变量。
We consider the graph coloring game, a game in which two players take turns properly coloring the vertices of a graph, with one player attempting to complete a proper coloring, and the other player attempting to prevent a proper coloring. We show that if a graph $G$ has a proper coloring in which the game coloring number of each bicolored subgraph is bounded, then the game chromatic number of $G$ is bounded. As a corollary to this result, we show that for two graphs $G_1$ and $G_2$ with bounded game coloring number, the Cartesian product $G_1 \square G_2$ has bounded game chromatic number, ans…
For a proper vertex coloring $c$ of a graph $G$, let $φ_c(G)$ denote the maximum, over all induced subgraphs $H$ of $G$, the difference between the chromatic number $χ(H)$ and the number of colors used by $c$ to color $H$. We define the chromatic discrepancy of a graph $G$, denoted by $φ(G)$, to be the minimum $φ_c(G)$, over all proper colorings $c$ of $G$. If $H$ is restricted to only connected induced subgraphs, we denote the corresponding parameter by $\hatφ(G)$. These parameters are aimed at studying graph colorings that use as few colors as possible in a graph and all its induced subgraph…
Suns in triangle-free graphs of large chromatic number
研究三角自由图中色数较大时是否存在诱导子图为特定太阳结构，证明色数至少为48的图存在特定太阳结构或近似结构。
The Gyárfás-Sumner conjecture says that for every forest $H$ and every integer $k$, if $G$ is $H$-free and does not contain a clique on $k$ vertices then it has bounded chromatic number. (A graph is $H$-free if it does not contain an induced copy of $H$.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest $H$, if $G$ is $H$-free and does not contain $K_{t,t}$ as a subgraph then it has bounded…
Problem of finding an optimal upper bound for the chromatic no. of a graph is still open and very hard. Borodin and Kostochka Conjecture is still open and if proved will improve Brook bound on Chromatic no. of a graph. Here we prove Borodin & Kostochka Conjecture for (1) (P4 Union K1)-free (2) P5-free (3) Chair-free graphs and 4) graphs with dense neighbourhoods. Certain known results follow as Corollaries.
Highly connected subgraphs with large chromatic number
本研究针对图论中高色数图的高连通子图存在性问题，改进了此前的上界结果，证明$g(k,m)$的更紧上界，向Thomassen的经典猜想推进了一步。
Let $G$ be a simple graph with maximum degree $Δ(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Δ(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $Δ(G)>n/3$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, Kühn and Osthus is affirmative.
A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on ver- tices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least positive integer m for which the graph is (m, k)-colourable. All triangle-free graphs on 8 or fewer vertices are (2, 1)-colourable. There are exactly four triangle-free graphs of order 9 which have 1-defective chromatic number 3. We show that these four graphs appear as subgraphs in almost all triangle-free graphs of order 10 with 1-defective chromatic numb…
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 $G$.
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 prove that any graph $G$ of minimum degree greater than $2k^2-1$ has a $(k+1)$-connected induced subgraph $H$ such that the number of vertices of $H$ that have neighbors outside of $H$ is at most $2k^2-1$. This generalizes a classical result of Mader, which states that a high minimum degree implies the existence of a highly connected subgraph. We give several variants of our result, and for each of these variants, we give asymptotics for the bounds. We also we compute optimal values for the case when $k=2$. Alon, Kleitman, Saks, Seymour, and Thomassen proved that in a graph of high chromati…
A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an odd colouring of G. Scott [Graphs and Combinatorics, 2001] proved that a graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph G is k-odd colourable if it can be partitioned into at most k odd induced subgraphs. We initiate the systematic study of odd colouring and odd chromatic n…
The robust chromatic number of graphs
本文系统研究图的1-鲁棒色数，定义为所有1-移除子图的最小色数，并类似考察其他标准图不变量。
For $k\geq 1$ and $n\geq 2k$, the Kneser graph $KG(n,k)$ has all $k$-element subsets of an $n$-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lovász that the chromatic number of $KG(n,k)$ is $n-2k+2$. Schrijver constructed a vertex-critical subgraph $SG(n,k)$ of $KG(n,k)$ with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges, however, no analogous construction is known except in trivial cases. We provide such a construction for $k=2$ and arbitrary $n\geq 4$ by means of a nice explicit combi…
Counting subgraphs of coloring graphs using shadow graphs
本文提出利用影子图色多项式显式构造着色图中诱导子图计数多项式的方法，给出超立方体情形公式，并以同色对多项式、不同色多项式的图对反驳相关猜想。
The celebrated Erdős--Stone--Simonovits theorem characterizes the asymptotic maximum edge density in $\mathcal{F}$-free graphs as $1 - 1/(χ(\mathcal{F})-1) + o(1)$, where $χ(\mathcal{F})$ is the minimum chromatic number of a graph in $\mathcal{F}$. In Examples 25 and 31 of [L. N. Coregliano and A. A. Razborov. Semantic limits of dense combinatorial objects. Uspekhi Mat. Nauk, 75(4(454)):45-152, 2020], it was shown that this result can be extended to the general setting of graphs with extra structure: the maximum asymptotic density of a graph with extra structure without some induced subgraphs …
The digraph chromatic number of a directed graph $D$, denoted $χ_A(D)$, is the minimum positive integer $k$ such that there exists a partition of the vertices of $D$ into $k$ disjoint sets, each of which induces an acyclic subgraph. For any $m \geq 1$, a digraph is weakly $m$-degenerate if each of its induced subgraphs has a vertex of in-degree or out-degree less than $m$. We introduce a generalization of the digraph chromatic number, namely $χ_m(D)$, which is the minimum number of sets into which the vertices of a digraph $D$ can be partitioned so that each set induces a weakly $m$-degenerate…
Acyclic subgraphs of digraphs with high chromatic number
研究了具有高色数的有向图的无环子图色数，证明了在特定条件下该色数下界为$n^{5/9-o(1)}$，并改进了现有界。
Locally bipartite subgraphs via multicolor Ramsey numbers
该研究针对Erdős-Hajnal猜想，改进了$f_4(k)$的上界，还证明了奇数$g\ge5$时对应函数$h_g(k)$的增长性，证实了Mohar和Wu2018年的相关猜想。
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