This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lovász in 1978, in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology. This conjecture stated that the \emph{Kneser graph} $\KG_{m,n}$, the graph with all $k$-element subsets of $\{1,2,...,n\}$ as vertices and all pairs of disjoint sets as edges, has chromatic number $n-2k+2$. Several other proofs have since been published (by Bárány, Schrijver, Dolnikov, Sarkaria, Kriz, Greene, and others), all of them based on some version of the Borsuk--Ulam theorem, but otherwise quite different. Each can be extended to yield some lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe that \emph{every} finite graph may be represented as a generalized Kneser graph, to which the above bounds apply.)
We show that these bounds are almost linearly ordered by strength, the strongest one being essentially Lovász' original bound in terms of a neighborhood complex. We also present and compare various definitions of a \emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lovász and of \kriz). A suitable box complex is equivalent to Lovász' complex, but the construction is simpler and functorial, mapping graphs with homomorphisms to $\Z_2$-spaces with $\Z_2$-maps.
The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K_2,G) and its suspension, respectively. These investigations follow the line of research initiated by Matousek and Ziegler who recognized a hierarchy of the different topological e…
Kierstead, Szemerédi, and Trotter showed that a graph with at most $\lfloor r/(2n)\rfloor^n$ vertices such that each ball of radius $r$ in it is $c$-colorable should have chromatic number at most $n(c-1)+1$. We show that this estimate is sharp in $r$. Namely, for every $n$, $r$, and $c$ we construct a graph $G$ containing $O((2rc)^{n-1}c)$ vertices such that $χ(G)\geq n(c-1)+1$, although each ball of radius $r$ in $G$ is $c$-colorable. The core idea is the construction of a graph whose neighborhood complex is homotopy equivalent to the join of neighborhood complexes of two given graphs.
There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conje…
We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying undirected graph can be arbitrarily large. We also investigate the minimum possible directed local chromatic number of oriented versions of ``topologically t-chromatic'' graphs. We show that this minimum for large enough t-chromatic Schrijver graphs and t-chromatic generalized Mycielski graphs of appropriate parameters is the upper integer part o…
We investigate group-theoretic "signatures" of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which …
The local chromatic number of a graph was introduced by Erdős et al. [4]. In [17] a connection to topological properties of (a box complex of) the graph was established and in [18] it was shown that if a graph is strongly topologically 4-chromatic then its local chromatic number is at least four. As a consequence one obtains a generalization of the following theorem of Youngs: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs'…
For studying topological obstructions to graph colorings, Hom-complexes were introduced by Lovász. A graph $T$ is called a test graph if for every graph $H$, the $k$-connectedness of $|Hom(T, H)|$ implies $χ(H)\geq k + 1 + χ(T)$. The proof of the famous Kneser conjecture is based on the fact that $\mathcal{K}_2$, the complete graph on $2$ vertices, is a test graph. This result was extended to all complete graphs by Babson and Kozlov. Their proof is based on generalized nerve lemma and discrete Morse theory. In this paper, we propose a new topological lower bound for the chromatic number of a s…
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…
The local chromatic number of a graph was introduced by Erdos et al. in 1986. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan which generalizes the Borsuk-Ulam theo…
In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers $n,k,r,s$ with $n\ge rk$ and $2\le s\le r$, the $r$-uniform general Kneser hypergraph $\mbox{KG}^r_s(n,k)$, has all $k$-subsets of $\{1,\dots,n\}$ as the vertex set and all multi-sets $\{A_1,\dots, A_r\}$ of $k$-subsets with $s$-wise empty intersections as the edge set. The case $r=s=2$, was considers by Kneser \cite{K} in 1955, where he conjectured that its chromatic number is $n-2(k-1)$. This was finally proved by Lovász \cite{L} in 1978. The case $r>…
We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if $H$ is a fixed planar graph that has a planar embedding with all the vertices with degree at least 4 on a single face, then graphs excluding $H$ as a topological minor have bounded nonrepetitive chromatic number. This is the largest class of graphs known to have bounded nonrepetitive chromatic number.
Kneser's 1955 conjecture -- proven by Lovász in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser's conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász' result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser's conjecture. Here we prove a common generalization of a…
Combinatorics, in particular graph theory, has a rich history of being a domain of successful applications of tools from other areas of mathematics, including topological methods. Here, we survey the study of the Hom-complexes, and the ways these can be used to obtain lower bounds for the chromatic numbers of graphs, presented in a recent series of papers \cite{BK03a,BK03b,BK03c,CK1,CK2,K4,K5}. The structural theory is developed and put in the historical context, culminating in the proof of the Lovász Conjecture, which can be stated as follows: For a graph G, such that the complex Hom(C_{2r+1}…
Lovasz's striking proof of Kneser's conjecture from 1978 using the Borsuk--Ulam theorem provides a lower bound on the chromatic number of a graph. We introduce the shore subdivision of simplicial complexes and use it to show an upper bound to this topological lower bound and to construct a strong Z_2-deformation retraction from the box complex (in the version introduced by Matousek and Ziegler) to the Lovasz complex. In the process, we analyze and clarify the combinatorics of the complexes involved and link their structure via several ``intermediate'' complexes.
Based on a newly proposed spatial data model - spatial chromatic model (SCM), we developed a spatial coding scheme, called full-coded ordinary arranged chromatic diagram (full-OACD). Full-OACD is a type of spatial tessellation, where space is partitioned into a number of subspaces such as cells, edges, and vertexes. These subspaces are called spatial particles and assigned with unique codes - chromatic codes. The generation, structures, computations, and properties of full-OACD are introduced and relations between chromatic codes and particle spatial topology are investigated, indicating that …
Mathematics Institute, University of Warwick（华威大学数学研究所）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极值组合与概率研究组）；Departament de Matemàtiques, Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学数学系）；School of Mathematics and Statistics, Ningbo University（宁波大学数学与统计学院）
We prove that the generating function for the symmetric chromatic polynomial of all connected graphs satisfies (after appropriate scaling change of variables) the Kadomtsev--Petviashvili integrable hierarchy of mathematical physics. Moreover, we describe a large family of polynomial graph invariants giving the same solution of the KP. In particular, we introduce the Abel polynomial for graphs and show this for its generating function. The key point here is a Hopf algebra structure on the space spanned by graphs and the behavior of the invariants on its primitive space.
Given $0<α\leqπ$, $ε>0$ and $n$, we define random graphs on the $d$-dimensional sphere by drawing $n$ i.i.d. uniform random points for the vertices, and edges $u {\sim} v$ whenever the geodesic distance between $u$ and $v$ is $ε$-close to $α$. This model generalizes distance graphs on spheres, and also random Borsuk graphs. Topological tools are known to give tight bounds for the chromatic number of Borsuk graphs. We now study the efficiency of one of these topological invariants, namely the connectivity of Lóvasz's neighborhood complex, to bound the chromatic number of this model of random gr…
Our purpose is to show that complements of line graphs enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal to a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph $\operatorname{KG}(n,2)$ that have a chromatic number equal to its chromatic number, namely $n-2$. In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the …
The DP-coloring problem is a generalization of the list-coloring problem in which the goal is to find an independent transversal in a certain topological cover of a graph $G$. In the online DP-coloring problem, the cover of $G$ is revealed one component at a time, and the independent transversal of the cover must be constructed in parts based on incomplete information. Kim, Kostochka, Li, and Zhu asked whether the chromatic numbers corresponding to these two graph coloring problems can have an arbitrarily large difference in a single graph. We answer this question in the affirmative by constru…
There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of $\mathbb{Z}_p$-Tucker lemma, Alishahi and Hajiabolhassan [{\it On the chromatic number of general Kneser hypergraphs, Journal of Combinatorial Theory, Series B, 2015}] introduced a lower bound for the chromatic number of Kneser hypergraphs ${\rm KG}^r({\mathcal H})$. Next, Meunier [{\it Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014}] improved their re…
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…
Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovasz, which invokes the Borsuk-Ulam theorem. Van Ngoc and Tuza used elementary combinatorial arguments to prove Stiebitz's theorem for 4-chromatic generalised Mycielski graphs, and asked if there is also an elementary combinatorial proof for higher chromatic number. We answer their question by showing that Stiebitz's theorem can be deduced from a version of Fan's combinatorial lemma. Our proof uses topological terminology, but is otherwise completely discrete and could be rewritten to avo…
In the way of proving Kneser's conjecture, László Lovász settled out a new lower bound for the chromatic number. He showed that if neighborhood complex $\mathcal{N}(G)$ of a graph $G$ is topologically $k$-connected, then its chromatic number is at least $k+3$. Then he completed his proof by showing that this bound is tight for the Kneser graph. However, there are some graphs where this bound is not useful at all, even comparing by the obvious bound; the clique number. For instance, if a graph contains no complete bipartite graph $\mathcal{K}_{l, m}$, then its Lovász's bound is at most $l+m-1$.…
A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive colourings with a bounded number of colours, thus proving a conjecture of Alon, Grytczuk, Haluszczak and Riordan (2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding a fixed minor, and graphs excluding a fixed topological minor.
We study a model of random graph where vertices are $n$ i.i.d. uniform random points on the unit sphere $S^d$ in $\mathbb{R}^{d+1}$, and a pair of vertices is connected if the Euclidean distance between them is at least $2- ε$. We are interested in the chromatic number of this graph as $n$ tends to infinity. It is not too hard to see that if $ε> 0$ is small and fixed, then the chromatic number is $d+2$ with high probability. We show that this holds even if $ε\to 0$ slowly enough. We quantify the rate at which $ε$ can tend to zero and still have the same chromatic number. The proof depends on c…
We introduce new methods for understanding the topology of $\Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $\Hom(T,-)$ and $\Hom(-,G)$ as functors from graphs to posets, and introduce a functor $(-)^1$ from posets to graphs obtained by taking atoms as vertices. Our main structural results establish useful interpretations of the equivariant homotopy type of $\Hom$ complexes in terms of spaces of equivariant poset maps and $Γ$-twisted products of spaces. When $P = F(X)$ is the face poset of a simplicial complex $…
Chromatic thresholds for linear equations and recurrence
Institute for Basic Science (IBS)（基础科学研究院）；Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；Department of Mathematics, Stanford University（斯坦福大学数学系）
To estimate the lower bound for the chromatic number of a graph $G$, Lovász associated a simplicial complex $\mathcal{N}(G)$ called the neighborhood complex and relates the topological connectivity of $\mathcal{N}(G)$ to the chromatic number of $G$. More generally he proved that the chromatic number of $G$ is bounded below by the topological connectivity of $\mathcal{N}(G)$ plus $3$. In this article, we consider the graphs of maximal degree at most $3$ and $4$-regular circulant graphs. We show that each connected component of the neighborhood complexes of these graphs is homotopy equivalent ei…
In 2011, Meunier conjectured that for positive integers $n,k,r,s$ with $ k\geq 2$, $r\geq 2$, and $n\geq \max (\{r,s\})k$, the chromatic number of $s$ -stable $r$-uniform Kneser hypergraphs is equal to $\left\lceil \frac{n-\max (\{r,s\})(k-1)}{r-1}\right\rceil $. It is a strengthened version of the conjecture proposed by Ziegler (2002), and Alon, Drewnowski and Łuczak (2009). The problem about the chromatic number of almost $s$-stable $r$ -uniform Kneser hypergraphs has also been introduced by Meunier (2011). For the $r=2$ case of the Meunier conjecture, Jonsson (2012) provided a purely combin…
Higher chromatic numbers $χ_s$ of simplicial complexes naturally generalize the chromatic number $χ_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $χ_s$ of $d$-complexes can become arbitrarily large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $χ_{d+1}=1$, and only little is known on $χ_s$ for $\lceil d/2\rceil<s\leq d$. A particular class of $d$-complexes are triangulations of $d$-manifolds. As a consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see …
In this paper, we find recursive formulas for the Tutte polynomial of a family of small-world networks: Farey graphs, which are modular and have an exponential degree hierarchy. Then, making use of these formulas, we determine the number of spanning trees, as well as the number of connected spanning subgraphs. Furthermore, we also derive exact expressions for the chromatic polynomial and the reliability polynomial of these graphs.
This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial χ(Q) of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chromatic polynomial. In particular, we show that Tutte's golden identity is a consequence of level-rank duality for SO(N) topological quantum field theories and Birman-Murakami-Wenzl algebras. This identity is a remarkable feature of the chromatic polynomial relating χ({ϕ+2}) for any …
Let $G$ be a non-trivial finite group. The well-known Dold's theorem states that: There is no continuous $G$-equivariant map from an $n$-connected simplicial $G$-complex to a free simplicial $G$-complex of dimension at most $n$. In this paper, we give a new generalization of Dold's theorem, by replacing "dimension at most $n$" with a sharper combinatorial parameter. Indeed, this parameter is the chromatic number of a new family of graphs, called strong compatibility graphs, associated to the target space. Moreover, in a series of examples, we will see that one can hope to infer much more infor…
Haviv ({\em European Journal of Combinatorics}, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over $\mathbb{R}$. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over $\mathbb{R}$ -- an important graph invariant from coding theory -- and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used i…
In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph $G=(V,E)$, an oriented $r$-coloring for $G$ is a partition of the vertex set $V$ into $r$ independent sets, such that all the arcs between two of these sets have the same direction. The oriented chromatic number of $G$ is the smallest integer $r$ such that $G$ permits an oriented $r$-coloring. In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs. Oriented co-graphs (short for oriented complement reducible graphs…
We introduce a generalization of the celebrated Lovász theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such as the relationship to Hoffman's ratio bound and to the chromatic number and study how they extend to higher dimensions. Like in the case of graphs, the higher dimensional theta number can be extended t…
In this work, we present a generalization of Gale's lemma. Using this generalization, we introduce two combinatorial sharp lower bounds for ${\rm conid}({\rm B}_0(G))+1$ and ${\rm conid}({\rm B}(G))+2$, two famous topological lower bounds for the chromatic number of a graph $G$.
We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, $R(G)$, and the topological Penrose polynomial, $P(G)$. The general framework provided by $Q(G)$ also leads to several other combinatorial interpretations these polynomials. In particular, we express $P(G)$, $R(G)$, and the Tutte polynomial, $T(G)$, as sums of chromatic polynomials of graphs derived from $G$; show that these polynomials count $k$-valuations of media…
