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.
We show that no tree on twenty vertices with maximum degree ten has Schur positive chromatic symmetric function, thereby providing a counterexample to a conjecture from the paper "Schur and e-positivity of trees and cut vertices".
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.
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 $\ell$.
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.
The main purpose of this note is to give a counterexample to the following conjecture, raised by Florian Frick [\textit{Int. Math. Res. Not. IMRN 2020 (13), 4037-4061 (2020)}]. Conjecture. Let $r\geq 3$ and let $\mathcal{F}$ be a set system. Then $$χ\left(\textrm{KG}^r\left(\mathcal{F}_{r-stab}\right)\right)\geq\left\lceil\frac{cd_r\left(\mathcal{F}\right)}{r-1}\right\rceil.$$
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…
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…
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 …
The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $χ(H)$ and $χ_l(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $χ_l (H) = χ(H)$. It is an interesting problem to find graphs that are chromatic-choosable. Kostochka and Woodall \cite{KW2001} conjectured that $χ_l(G^2) = χ(G^2)$ for every graph $G$, which is called List Square Coloring Conjecture. In this paper, we give infinitely man…
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.
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 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 …
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.
Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of $n$-well-quasi-order introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not $2$-well-quasi-ordered. This counterexa…
Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half its vertices has maximum degree at least $\sqrt{d}$, where $d$ is the valency of the hypercube. This was generalised by Alon and Zheng who proved that every Cayley graph on an elementary abelian $2$-group has the same property. Very recently, Potechin and Tsang proved an analogous results for Cayley graphs on abelian groups. They also conjectured that all Cayley graphs have the analogous property. We disprove thi…
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…
The Hall ratio of a graph $G$ is the maximum value of $v(H) / α(H)$ taken over all non-null subgraphs $H$ of $G$. For any graph, the Hall ratio is a lower-bound on its fractional chromatic number. In this note, we present various constructions of graphs whose fractional chromatic number grows much faster than their Hall ratio. This refutes a conjecture of Harris.
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…
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.
We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.
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…
In this paper, we present a minimal counterexample to a conjecture of Perles that answers a question of Haase and Ziegler. The example is a simple 4-polytope that has an induced 3-connected 3-regular subgraph, whose graph complement is connected. This subgraph is planar and not the graph of a facet of the polytope.
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
University of West Bohemia（西波希米亚大学）；CNRS（法国国家科学研究中心）；IRIF（IRIF实验室）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Information Studies in Novo Mesto（诺沃梅斯托信息研究学院）；Rudolfovo – Science and Technology Centre（鲁多维福科学与技术中心）；University of Southern Denmark（南丹麦大学）
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…
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…
Hedetniemi conjectured in 1966 that $χ(G \times H) = \min\{χ(G), χ(H)\}$ for all graphs $G$ and $H$. Here $G\times H$ is the graph with vertex set $ V(G)\times V(H)$ defined by putting $(x,y)$ and $(x',y')$ adjacent if and only if $xx'\in E(G)$ and $yy'\in E(H)$. This conjecture received a lot of attention in the past half century. Recently, Shitov refuted this conjecture. Let $p$ be the minimum number of vertices in a graph of odd girth $7$ and fractional chromatic number greater than $3+4/(p-1)$. Shitov's proof shows that Hedetniemi's conjecture fails for some graphs with chromatic number ab…
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.
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…
