We show Péter Csorba's conjecture that the graph homomorphism complex Hom(C_5,K_{n+2}) is homeomorphic to a Stiefel manifold, the space of unit tangent vectors to the n-dimensional sphere. For this a general tool is developed that allows to replace the complexes Hom(G, K_n) by smaller complexes that are homeomorphic to them whenever G is a graph for which those complexes are manifolds. The equivariant version of Csorba's conjecture is proved up to homotopy.
We also study certain subdivisions of simplicial manifolds that are related to the interval poset of their face posets and their connection with geometric approximations to diagonal maps.
We show that the n-th power of the first Stiefel-Whitney class of the Z_2-operation on the graph complex Hom(C_{2r+1},K_{n+2})$ is zero, confirming a conjecture by Babson and Kozlov. This proves the strong form of their graph colouring theorem, which they had only proven for odd n. Our proof is also considerably simpler than their proof of the weak form of the theorem, which is also known as the Lovász conjecture.
Several possible definitions of local injectivity for a homomorphism of an oriented graph $G$ to an oriented graph $H$ are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when $G$ is given and $H$ is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.
Each of several possible definitions of local injectivity for a homomorphism of an oriented graph $G$ to an oriented graph $H$ leads to an injective oriented colouring problem. For each case in which such a problem is solvable in polynomial time, we identify a set $\mathcal{F}$ of oriented graphs such that an oriented graph $G$ has an injective oriented colouring with the given number of colours if and only if there is no $F \in \mathcal{F}$ for which there is a locally-injective homomorphism of $F$ to $G$.
Given a graph $G$, the $k$-mixing problem asks: Can one obtain all $k$-colourings of $G$, starting from one $k$-colouring $f$, by changing the colour of only one vertex at a time, while at each step maintaining a $k$-colouring? More generally, for a graph $H$, the $H$-mixing problem asks: Can one obtain all homomorphisms $G \to H$, starting from one homomorphism $f$, by changing the image of only one vertex at a time, while at each step maintaining a homomorphism $G \to H$? This paper focuses on a generalization of $k$-colourings, namely $(p,q)$-circular colourings. We show that when $2 < \fra…
For a fixed graph $H$, the reconfiguration problem for $H$-colourings (i.e. homomorphisms to $H$) asks: given a graph $G$ and two $H$-colourings $φ$ and $ψ$ of $G$, does there exist a sequence $f_0,\dots,f_m$ of $H$-colourings such that $f_0=φ$, $f_m=ψ$ and $f_i(u)f_{i+1}(v)\in E(H)$ for every $0\leq i<m$ and $uv\in E(G)$? If the graph $G$ is loop-free, then this is the equivalent to asking whether it possible to transform $φ$ into $ψ$ by changing the colour of one vertex at a time such that all intermediate mappings are $H$-colourings. In the affirmative, we say that $φ$ reconfigures to $ψ$. …
A tropical graph $(H,c)$ consists of a graph $H$ and a (not necessarily proper) vertex-colouring $c$ of $H$. Given two tropical graphs $(G,c_1)$ and $(H,c)$, a homomorphism of $(G,c_1)$ to $(H,c)$ is a standard graph homomorphism of $G$ to $H$ that also preserves the vertex-colours. We initiate the study of the computational complexity of tropical graph homomorphism problems. We consider two settings. First, when the tropical graph $(H,c)$ is fixed; this is a problem called $(H,c)$-COLOURING. Second, when the colouring of $H$ is part of the input; the associated decision problem is called $H$-…
This work brings together ideas of mixing graph colourings, discrete homotopy, and precolouring extension. A particular focus is circular colourings. We prove that all the $(k,q)$-colourings of a graph $G$ can be obtained by successively recolouring a single vertex provided $k/q\geq 2col(G)$ along the lines of Cereceda, van den Heuvel and Johnson's result for $k$-colourings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colourings coincide. As a corollary, we obtain an Albertson-type extension theorem for…
A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The H-Colouring problem is to decide whether or not a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the Surjective H-Colouring problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exac…
By Lovasz' proof of the Kneser conjecture, the chromatic number of a graph G is bounded from below by the index of the Z_2-space Hom(K_2,G) plus two. We show that the cohomological index of Hom(K_2,G) is also greater than the cohomological index of the Z_2-space Hom(C_{2r+1}, G) for r>0. This gives a new and simple proof of the strong form of the graph colouring theorem by Babson and Kozlov, which had been conjectured by Lovasz, and at the same time shows that it never gives a stronger bound than can be obtained by Hom(K_2, G). The proof extends ideas introduced by Zivaljevic in a previous ele…
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}…
For graphs $G$ and $H$, an $H$-colouring of $G$ is a map $ψ:V(G)\rightarrow V(H)$ such that $ij\in E(G)\Rightarrowψ(i)ψ(j)\in E(H)$. The number of $H$-colourings of $G$ is denoted by $\hom(G,H)$. We prove the following: for all graphs $H$ and $δ\geq3$, there is a constant $κ(δ,H)$ such that, if $n\geqκ(δ,H)$, the graph $K_{δ,n-δ}$ maximises the number of $H$-colourings among all connected graphs with $n$ vertices and minimum degree $δ$. This answers a question of Engbers. We also disprove a conjecture of Engbers on the graph $G$ that maximises the number of $H$-colourings when the assumption o…
Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invariants can be computed purely combinatorially by enumerating colourings on the edges of a triangulation T. These edge colourings can be interpreted as embeddings of surfaces in T. We give a characterisation of how these embedded surfaces intersect with the tetrahedra of T. This is done by characterising isotopy classes of simple closed loops in the 3-punctured disk. As a direct result we obtain a new system of coordinate…
A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G which embeds as a quadrangulation in the real projective space P^n has chromatic number n+2 or higher, unless G is bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996), 219-227]. The family of quadrangulations of projective spaces includes all complete graphs, all Mycielski graphs, and certain graphs homomorphic to Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser theorem.
The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H …
Colouring negative exact-distance graphs of signed graphs
Zhejiang Normal University（浙江师范大学）；Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息学研究所）；Centre d’Analyse et de Mathématiques Sociales (CNRS, UMR 8557)（分析与数学社会中心）；Instituto de Ingeniería Matemática-CIMFAV（数学工程研究所-CIMFAV）；Universidad de Valparaíso（瓦尔帕莱索大学）；Computer Science Institute of Charles University（查理大学计算机科学研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）
For graphs $G$ and $H$, an {\em $H$-colouring} of $G$ (or {\em homomorphism} from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colourings generalize such graph theory notions as proper colourings and independent sets. For a given $H$, $k \in V(H)$ and $G$ we consider the proportion of vertices of $G$ that get mapped to $k$ in a uniformly chosen $H$-colouring of $G$. Our main result concerns this quantity when $G$ is regular and bipartite. We find numbers $0 \leq a^-(k) \leq a^+(k) \leq 1$ with the property that for all such $G$, with h…
We present a necessary and sufficient condition for a graph of odd-girth $2k+1$ to bound the class of $K_4$-minor-free graphs of odd-girth (at least) $2k+1$, that is, to admit a homomorphism from any such $K_4$-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every $K_4$-minor free graph of odd-girth $2k+1$ admits a homomorphism to the projective hypercube of dimension $2k$. This supports a conjecture of the third author which generalizes the four-color theorem and relates to several outstanding conjectures such as Se…
A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. Let $H$ be a multigraph with the property that any two distinct vertices share at most one common neighbour. We show that the generating function for (list) graph homomorphisms from $G$ to $H$ can be calculated in subexponential time $2^{O\left(\sqrt{tn\log(n)}\right)}$ for $n=|V(G)|$ in the class of $P_t$-free graphs $G$. As a corollary, we show that the number of 3-colourings of a $P_t$-free graph $G$ can be found in subexponential time. On the other hand, no subexponential time algorithm exi…
We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovasz. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds provide examples of highly connected, highly symmetric manifolds.
The complexity of {\sc Colouring} is fully understood for $H$-free graphs, but there are still major complexity gaps if two induced subgraphs $H_1$ and $H_2$ are forbidden. Let $H_1$ be the $s$-vertex cycle $C_s$ and $H_2$ be the $t$-vertex path $P_t$. We show that {\sc Colouring} is polynomial-time solvable for $s=4$ and $t\leq 6$, strengthening several known results. Our main approach is to initiate a study into the boundedness of the clique-width of atoms (graphs with no clique cutset) of a hereditary graph class. We first show that the classifications of boundedness of clique-width of $H$-…
In this paper, we first discuss the topological properties of projective Stiefel manifolds, we compute their cohomology rings and classify their cohomology endomorphisms; Then by embedding the flag manifold of a classical Lie group into its corresponding infinite dimensional projective Stiefel manifold(which is homotopic to the product of infinite dimensional complex projective space $\mathbb{C}P^{\infty}$), we define the Schubert polynomials and dual Schubert polynomials. Finally we discuss the property and the computation of these polynomials.
A fractional colouring of a graph $G$ is a function that assigns a non-negative real value to all possible colour-classes of $G$ containing any vertex of $G$, such that the sum of these values is at least one for each vertex. The fractional chromatic number is the minimum sum of the values assigned by a fractional colouring over all possible such colourings of $G$. Introduced by Bosica and Tardif, fractional covers are an extension of fractional colourings whereby the real-valued function acts on all possible subgraphs of $G$ belonging to a given class of graphs. The fractional chromatic numbe…
The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in logspace or is hard for NL. More precisely, we introduce a digraph structure we call a circular N, and prove the following dichotomy: if H contains no circular N then LHOM(H) admits a logspace algorithm, and otherwise LHOM(H) is h…
Given a homogeneous ideal $I \subseteq k[x_0,\dots,x_n]$, the Containment problem studies the relation between symbolic and regular powers of $I$, that is, it asks for which pair $m, r \in \mathbb{N}$, $I^{(m)} \subseteq I^r$ holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne -- Huneke Conjecture and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in $\mathb…
A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring, or equivalently, $t_H(W)+t_H(1-W)\geq 2^{1-e(H)}$ holds for every graphon $W:[0,1]^2\rightarrow [0,1]$, where $t_H(.)$ denotes the homomorphism density of the graph $H$. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989). We prove a graph homomorphism inequality that extends the commonality of paths and cycl…
For any two graphs $G$ and $H$ Lovász has defined a cell complex $Hom(G,H)$ having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with $G$ a cycle of odd length. More specifically, we show that: if $Hom(C_{2r+1},G)$ is $k$-connected, then $χ(G)\geq k+4$. Our actual statement is somewhat sharper, as we find obstructions already in the non-vanishing of powers of certain Stiefel-Whitney classes.
We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus $\mathbb{Z}_m^n$, where $m$ is even, to any fixed graph: we show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some dominant phase. This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of pr…
To any two graphs G and H one can associate a cell complex Hom(G,H) by taking all graph multihomorphisms from G to H as cells. In this paper we prove the Lovasz Conjecture which states that if Hom(C_{2r+1},G) is k-connected, then χ(G)\geq k+4, where r,k\in Z, r\geq 1, k\geq -1, and C_{2r+1} denotes the cycle with 2r+1 vertices. The proof requires analysis of the complexes Hom(C_{2r+1},K_n). For even n, the obstructions to graph colorings are provided by the presence of torsion in H^*(Hom(C_{2r+1},K_n);Z). For odd n, the obstructions are expressed as vanishing of certain powers of Stiefel-Whitn…
Using the definition of colouring of $2$-edge-coloured graphs derived from $2$-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to $2$-edge-coloured graphs. We find closed forms for the first three coefficients of this polynomial that generalize the known results for the chromatic polynomial of a graph. We classify those $2$-edge-coloured graphs that have a chromatic polynomial equal to the chromatic polynomial of the underlying graph, when every vertex is incident to edges of both colours. Finally, we examine the behaviour of the roots of this polynomial, hig…
$Hom(G,H)$ is a polyhedral complex defined for any two undirected graphs $G$ and $H$. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that $Hom(K_m,K_n)$ is homotopy equivalent to a wedge of $(n-m)$-dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph $G$, and integers $m\geq 2$ and $k\geq -1$, we have $\varpi_1^k(\thom(K_m,G))\neq 0$, then $χ(G)\geq k+m$; here $Z_…
