In this paper we introduce the notion of $Σ$-colouring of a graph $G$: For given subsets $Σ(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that appear together in some $Σ(v)$ receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptotic versions of Wegner's and Borodin's Conjecture on the planar version of these two colourings. Using a recent approach of Havet et al., we reduce the problem to edge-colouring of multigraphs, and then use Kahn's result that the list chromatic index is close to the fractional chromatic index.
Our results are based on a strong structural lemma for graphs embeddable in a fixed surface, which also implies that the size of a clique in the square of a graph of maximum degree $Δ$ embeddable in some fixed surface is at most $\frac32\,Δ$ plus a constant.
For any two non-negative integers h and k, h > k, an L(h, k)-colouring of a graph G is a colouring of vertices such that adjacent vertices admit colours that at least differ by h and vertices that are two distances apart admit colours that at least differ by k. The smallest positive integer δ such that G permits an L(h, k)-colouring with maximum colour δ is known as the L(h, k)-chromatic number (L(h, k)-colouring number) denoted by λ_{h,k}(G). In this paper, we discuss some interesting invariants in hypergraphs. In fact, we study the relation between the spectral gap and L(2, 1)-chromatic numb…
We consider distance colourings in graphs of maximum degree at most $d$ and how excluding one fixed cycle length $\ell$ affects the number of colours required as $d\to\infty$. For vertex-colouring and $t\ge 1$, if any two distinct vertices connected by a path of at most $t$ edges are required to be coloured differently, then a reduction by a logarithmic (in $d$) factor against the trivial bound $O(d^t)$ can be obtained by excluding an odd cycle length $\ell \ge 3t$ if $t$ is odd or by excluding an even cycle length $\ell \ge 2t+2$. For edge-colouring and $t\ge 2$, if any two distinct edges con…
A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance two must receive distinct integers. The lambda chromatic number (or the $λ$ number) of a graph $G$ is the least positive integer among all the maximum assigned positive integer over all possible lambda colouring of the graph $G$. Here we have primarily shown that every graph with lambda chromatic number $t$ can be embedded in a graph, with lambda chromatic n…
In this paper we prove that if an infinite circulant graph with $k$ distances has a perfect $2$-colouring with parameters $(b, c)$, then $b + c \leq 2k + \frac{b+c}{q^t}$ for all positive integers $t$ and primes $q$ satisfying $\frac{b+c}{gcd(b,c)}\vdots q^t$. In addition, we show that if $b + c = q^t$, then this necessary condition becomes sufficient for the existence of perfect $2$-colourings in circulant graphs.
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）
The foldings of a connected graph $G$ are defined as follows. First, $G$ is a folding of itself. Let $G'$ be a graph obtained from $G$ by identifying two vertices at distance 2 in $G$. Then every folding of $G'$ is a folding of $G$. The folding number of $G$ is the minimum order of a complete folding of $G$. Theorem: The folding number of every graph equals its chromatic number.
Given a multigraph $G$ and a positive integer $t$, the distance-$t$ chromatic index of $G$ is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than $t$ edges must receive different colours. Let $π'_t(d)$ and $τ'_t(d)$ be the largest values of this parameter over the class of planar multigraphs and of (simple) trees, respectively, of maximum degree $d$. We have that $π'_t(d)$ is at most and at least a non-trivial constant multiple larger than $τ'_t(d)$. (We conjecture $\limsup_{d\to\infty}π'_2(d)/τ'_2(d) =9/4$ in p…
We present results referring to the Hadwiger-Nelson problem which asks for the minimum number of colours needed to colour the plane with no two points at distance $1$ having the same colour. Exoo considered a more general problem concerning graphs $G_{[a,b]}$ with $\mathbb{R}^2$ as the vertex set and two vertices adjacent if their distance is in the interval $[a,b]$. Exoo conjectured $χ(G_{[a,b]}) = 7$ for sufficiently small but positive difference between $a$ and $b$. We partially answer this conjecture by proving that $χ(G_{[a,b]}) \geq 5$ for $b > a$. A $j$-fold colouring of graph $G = (V,E…
One of the most famous applications of Graph Theory is in the field of Channel Assignment Problems. There are varieties of graph colouring concepts that are used for different requirements of frequency assignments in communication channels. We introduce here L(t, 1)-colouring of graphs. This has its foundation in T-colouring and L(p, q)-colouring. For a given finite set T including zero, an L(t, 1)-colouring of a graph G is an assignment of non-negative integers to the vertices of G such that the difference between the colours of adjacent vertices must not belong to the set T and the colours o…
We characterise which simplicial surfaces can be folded onto a triangle. We define a notion of folding that incorporates the non-intersection-properties of real materials. All of the surfaces foldable onto a triangle admit a vertex-3-colouring. Based on this colouring, we can describe the surface by three involutions that act on the faces of the surface. A simplicial surface is foldable onto a triangle if and only if there exists a cyclic permutation on all faces, whose products with the involutions have a specified number of cycles. In addition, we show that all simplicial surfaces that can b…
A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobás-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentio…
A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given $n$ points $x_1, \ldots,x_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $x_i$ and $x_j$ is at most $r$. We investigate the clique chromatic number of such graphs. We first show that the clique chromatic number is at most 9 for any geom…
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.
Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerp…
Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no $K_{t+1}$-minor can be $t$-coloured so that each colour class induces a subgraph of bounded maximum degree. We prove a common generalisation of these theorems with a weaker assumption about excluded subgraphs. This result leads to new defective colouring results for several graph classes, including graphs with linear crossing number, graphs with given thickness (with r…
The Total Colouring Conjecture suggests that $Δ+3$ colours ought to suffice in order to provide a proper total colouring of every graph $G$ with maximum degree $Δ$. Thus far this has been confirmed up to an additive constant factor, and the same holds even if one additionally requires every pair of neighbours in $G$ to differ with respect to the sets of their incident colours, so called pallets. Within this paper we conjecture that an upper bound of the form $Δ+const.$ still remains valid even after extending the distinction requirement to pallets associated with vertices at distance at most $…
In this paper, we study orthogonal colourings of random geometric graphs. Two colourings of a graph are orthogonal if they have the property that when two vertices receive the same colour in one colouring, then those vertices receive distinct colours in the other colouring. A random geometric graph $RG(n,r)$ is a graph constructed by randomly placing $n$ vertices in the unit square and connecting two vertices with an edge if and only if their distance is less than the threshold $r$. We show first that random geometric graphs with $r>n^{-α}$, where $0\leq α\leq\frac{1}{4}$, have an orthogonal c…
We study the computational complexity of distance games, a class of combinatorial games played on graphs. A move consists of colouring an uncoloured vertex subject to it not being at certain distances determined by two sets, D and S. D is the set of forbidden distances for colouring vertices in different colors, while S is the set of forbidden distances for the same colour. The last player to move wins. Well-known examples of distance games are Node-Kayles, Snort, and Col, whose complexities were shown to be PSPACE-hard. We show that many more distance games are also PSPACE-hard.
Let $C_n^k$ be the $k$-th power of a cycle on $n$ vertices (i.e. the vertices of $C_n^k$ are those of the $n$-cycle, and two vertices are connected by an edge if their distance along the cycle is at most $k$). For each vertex draw uniformly at random a subset of size $c$ from a base set $S$ of size $s=s(n)$. In this paper we solve the problem of determining the asymptotic probability of the existence of a proper colouring from the lists for all fixed values of $c,k$, and growing $n$.
We consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$. We obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours necessary for such a colouring. One is a bound of $(2-\eps)Δ^t$ for graphs of maximum degree at most $Δ$, where $\eps$ is some absolute positive constant independent of $t$. The other is a bound of $O(Δ^t/\log Δ)$ (as $Δ\to\infty$) for graphs of maximum degree at most $Δ$ and girth at least $2t+1$. The first bound is an analogue of Molloy and Reed's bound on t…
Let $F$ be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph $G$ with no bichromatic subgraph in $F$ is $\F$-free. The $F$-free chromatic number $χ(G,F)$ of a graph $G$ is the minimum number of colours in an $F$-free colouring of $G$. For appropriate choices of $F$, several well-known types of colourings fit into this framework, including acyclic colourings, star colourings, and distance-2 colourings. This paper studies $F$-free colourings of the cartesian product of graphs. Let $H$ be the cartesian product of the graphs $G_1,G_2,..…
The packing chromatic number $χ$ $ρ$ (G) of a graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V 1 ,. .. , V k , in such a way that every two distinct vertices in V i are at distance greater than i in G for every i, 1 $\le$ i $\le$ k. Recently, Balogh, Kostochka and Liu proved that $χ$ $ρ$ is not bounded in the class of subcubic graphs [Packing chromatic number of subcubic graphs, Discrete Math. 341 (2018), 474483], thus answering a question previously addressed in several papers. However, several subclasses of cubic or subcubic g…
Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a proper total colouring of a graph $G=(V,E)$ with maximum degree $Δ$. We say vertices $u,v\in V$ are sum distinguished if $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. By $χ"_{Σ,r}(G)$ we denote the least integer $k$ admitting such a colouring $c$ for which every $u,v\in V$, $u\neq v$, at distance at most $r$ from each other are sum distinguished in $G$. For every positive integer $r$ an infinite family of examples is known with $χ"_{Σ,r}(G)=Ω(Δ^{r-1})$. In this paper we prove that $χ"_{Σ,r}(G)\leq (2+o(1))Δ^{r-1}$ for every integer $r\geq 3$ and e…
Fix a palette $\mathcal K$ of $Δ+1$ colours, a graph with maximum degree $Δ$, and a subset $M$ of the edge set with minimum distance between edges at least $9$. If the edges of $M$ are arbitrarily precoloured from $\mathcal K$, then there is guaranteed to be a proper edge-colouring using only colours from $\mathcal K$ that extends the precolouring on $M$ to the entire graph. This result is a first general precolouring extension form of Vizing's theorem, and it proves a conjecture of Albertson and Moore under a slightly stronger distance requirement. We also show that the condition on the dista…
Johann Radon Institute for Computational and Applied Mathematics (RICAM)（约翰·拉德诺计算与应用数学研究所）；Austrian Academy of Sciences（奥地利科学院）；Johannes Kepler University Linz（林茨约翰内斯·开普勒大学）；Research Institute for Symbolic Computation (RISC)（符号计算研究所）；Czech Technical University in Prague（布拉格捷克理工大学）
We study the existence of an $n$-fold rotationally symmetric placement of a symmetric graph in the plane allowing a continuous deformation that preserves the symmetry and the distances between adjacent vertices. We show that such a flexible placement exists if and only if the graph has a NAC-colouring satisfying an additional property on the symmetry; a NAC-colouring is a surjective edge colouring by two colours such that every cycle is either monochromatic, or there are at least two edges of each colour.
Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $Δ$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours, i.e. $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. The least integer $k$ admitting such colouring $c$ under which every $u,v\in V$ at distance $1\leq d(u,v)\leq r$ in $G$ are sum distinguished is denoted by ${\rm ts}_r(G)$. Such graph invariants link the concept of the total vertex irregularity strength of graphs with so called 1-2-Conjecture, whose conce…
Let $G$ be the unit distance graph in the plane. A well-known problem in combinatorial geometry is that of determining the chromatic number of $G$. It is known that $4\le χ(G)\le 7$. The upper bound of 7 is obtained using tilings of the plane. The present paper studies two problems where we seek proper colourings of $G$, adding restrictions inspired by tilings: Let $H(ε)$ be the graph whose vertices are the points of ${\mathbb R}^2$, with an edge between two points if their distance lies in the interval $[1,1+ε]$. We show that for small $ε$, $0<ε\le \frac{3\sqrt{2}}{4}-1$, we have $6\le χ(H(ε)…
An $L(2,1)$-labelling of a finite graph $Γ$ is a function that assigns integer values to the vertices $V(Γ)$ of $Γ$ (colouring of $V(Γ)$ by ${\mathbb{Z}}$) so that the absolute difference of two such values is at least $2$ for adjacent vertices and is at least $1$ for vertices which are precisely distance $2$ apart. The lambda number $λ(Γ)$ of $Γ$ measures the least number of integers needed for such a labelling (colouring). A power graph $Γ_G$ of a finite group $G$ is a graph with vertex set as the elements of $G$ and two vertices are joined by an edge if and only if one of them is a positive…
Consider the graph $\mathbb{H}(d)$ whose vertex set is the hyperbolic plane, where two points are connected with an edge when their distance is equal to some $d>0$. Asking for the chromatic number of this graph is the hyperbolic analogue to the famous Hadwiger-Nelson problem about colouring the points of the Euclidean plane so that points at distance $1$ receive different colours. As in the Euclidean case, one can lower bound the chromatic number of $\mathbb{H}(d)$ by $4$ for all $d$. Using spectral methods, we prove that if the colour classes are measurable, then at least $6$ colours are are …
We improve the current best bound for distinct distances on non-ruled algebraic surfaces in ${\mathbb R}^3$. In particular, we show that $n$ points on such a surface span $Ω\left(n^{32/39-\varepsilon}\right)$ distinct distances, for any $\varepsilon>0$. Our proof adapts the proof of Székely for the planar case, which is based on the crossing lemma. As part of our proof for distinct distances on surfaces, we also obtain new results for distinct distances between circles in ${\mathbb R}^3$. Consider point sets ${\mathcal P}_1$ and ${\mathcal P}_2$ of respective sizes $m$ and $n$, such that each …
KU Leuven（荷语鲁汶大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉德纳计算与应用数学研究所）；Faculty of Information Technology, Czech Technical University in Prague（布拉格捷克理工大学信息技术学院）；School of Mathematical Sciences, Lancaster University, UK（兰开斯特大学数学科学学院）
A recent "hidden symmetry" conjecture of B. Gin-ge Chen et al is resolved, concerning the dimension of the mechanical modes of a generic $2$-periodic triangulated surface $O$ in $R^3$ whose structure graph corresponds to a triangular tiling of $R^2$. We introduce an indexing of the terms in an associated sparse determinant $p_O(z_1, z_2)$ by means of oriented $3$-colourings of the underlying sparse graph, and use this and vertex splitting arguments to show that $p_O(z_1,z_2)$ is palindromic or antipalindromic, up to a shift index. This implies the conjectured dimension $1$ phenomenon for these…
In this paper, we introduce the notion of "geodesic cover" for Fuchsian groups, which summons copies of fundamental polygons in the hyperbolic plane to cover pairs of representatives realizing distances in the corresponding hyperbolic surface. Then we use estimates of geodesic-covering numbers to study the distinct distances problem in hyperbolic surfaces. Especially, for $Y$ from a large class of hyperbolic surfaces, we establish the nearly optimal bound $\geq c(Y)N/\log N$ for distinct distances determined by any $N$ points in $Y$, where $c(Y)>0$ is some constant depending only on $Y$. In pa…
We investigate a type of distance between triangulations on finite type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism and our main results are upper bounds on distance between triangulations that only depend on the topology of the surface.
For a knot $K$, Kakimizu introduced a simplicial complex whose vertices are all the isotopy classes of minimal genus spanning surfaces for $K$. The first purpose of this paper is to prove the 1-skeleton of this complex has diameter bounded by a function quadratic in knot genus, whenever $K$ is atoroidal. The second purpose of this paper is to prove the intersection number of two minimal genus spanning surfaces for $K$ is also bounded by a function quadratic in knot genus, whenever $K$ is atoroidal. As one application, we prove the simple connectivity of Kakimizu's complex among all atoroidal g…
