We show that for loopless $6$-regular triangulations on the torus the gap between the choice number and chromatic number is at most $2$. We also show that the largest gap for graphs embeddable in an orientable surface of genus $g$ is of the order $Θ(\sqrt{g})$, and moreover for graphs with chromatic number of the order $o(\sqrt{g}/\log_{2}(g))$ the largest gap is of the order $o(\sqrt{g})$.
This paper extends the scenario of the Four Color Theorem in the following way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly) embedded into R^d. We investigate lower and upper bounds on the maximum (weak and strong) chromatic number of hypergraphs in H(d,k). For example, we can prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak chromatic number is Omega(log n/log log n), whereas the weak chromatic number for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.
Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We show that the number of vertex-labelled cubic multigraphs embeddable on $\mathbb{S}_g$ with $2n$ vertices is asymptotically $c_g n^{5(g-1)/2-1}γ^{2n}(2n)!$, where $γ$ is an algebraic constant and $c_g$ is a constant depending only on the genus $g$. We also derive an analogous result for simple cubic graphs and weighted cubic multigraphs. Additionally we prove that a typical cubic multigraph embeddable on $\mathbb{S}_g$, $g\ge 1$, has exactly one non-planar component.
The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As …
Given a function $g=g(n)$ we let ${\mathcal E}^g$ be the class of all graphs $G$ such that if $G$ has order $n$ (that is, has $n$ vertices) then it is embeddable in some surface of Euler genus at most $g(n)$, and let ${\widetilde{\mathcal E}}^g$ be the corresponding class of unlabelled graphs. We give estimates of the sizes of these classes. For example we show that if $g(n)=o(n/\log^3n)$ then the class ${\mathcal E}^{g}$ has growth constant $γ_{\mathcal P}$, the (labelled) planar graph growth constant; and when $g(n) = O(n)$ we estimate the number of n-vertex graphs in ${\mathcal E}^{g}$ and …
Given a `genus' function $g=g(n)$, we let $\mathcal{E}^g$ be the class of all graphs $G$ such that if $G$ has order $n$ (that is, has $n$ vertices) then it is embeddable in a surface of Euler genus at most $g(n)$. Let the random graph $R_n$ be sampled uniformly from the graphs in $\mathcal{E}^g$ on vertex set $[n]=\{1,\ldots,n\}$. Observe that if $g(n)$ is 0 then $R_n$ is a random planar graph, and if $g(n)$ is sufficiently large then $R_n$ is a binomial random graph $G(n,\tfrac12)$. We investigate typical properties of $R_n$. We find that for \emph{every} genus function $g$, with high probabi…
A method is suggested for construction of quadrangulations of the closed orientable surface with given genus g and either (1) with given chromatic number or (2) with given order allowed by the genus g. In particular, N. Hartsfield and G. Ringel's results [Minimal quadrangulations of orientable surfaces, J. Combin. Theory, Series B 46 (1989) 84-95] are generalized by way of generating new minimal quadrangulations of infinitely many other genera.
In this paper, by developing several new adjacency lemmas about a path on $4$ or $5$ vertices, we show that the average degree of 7-critical graphs is at least 6. It implies Vizing's planar graph conjecture for planar graphs with maximum degree $7$ and its extension to graphs embeddable in a surface with nonnegative Euler characteristic due to Sanders and Zhao (J. Combin. Theory Ser. B 83 (2001) 201-212 and J. Combin. Theory Ser. B 87 (2003) 254-263) and Zhang (Graphs and Combinatorics 16 (2000) 467-495).
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. …
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 …
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'…
On colorings of hypergraphs embeddable in $\mathbb{R}^d$
Keimyung University（启明大学）；Einstein Institute of Mathematics, Hebrew University（希伯来大学爱因斯坦数学研究所）；Mathematics Research Institute, Universidad de Valladolid（瓦拉多利德大学数学研究所）；CMSA, Harvard University（哈佛大学数学与统计应用中心）
The clustered chromatic number of a graph class is the minimum integer $t$ such that for some $C$ the vertices of every graph in the class can be colored in $t$ colors so that every monochromatic component has size at most $C$. We show that the clustered chromatic number of the class of graphs embeddable on a given surface is four, proving the conjecture of Esperet and Ochem. Additionally, we study the list version of the concept and characterize the minor-closed classes of graphs of bounded treewidth with given clustered list chromatic number. We further strengthen the above results to solve …
A pseudocircle is a simple closed curve on some surface; an arrangement of pseudocircles is a collection of pseudocircles that pairwise intersect in exactly two points, at which they cross. Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size at most four are embeddable into the sphere, and asked if an analogous result holds for embeddability into orientable surfaces of higher genus. We answer this question positively: An arrangement of pseudocircles is embeddable into an orientable surface of genus~$g$ if and only i…
We consider the thickness $θ(G))$ and outerthickness $θ_o(G)$ of a graph G in terms of its orientable and nonorientable genus. Dean and Hutchinson provided upper bounds for thickness of graphs in terms of their orientable genus. More recently, Concalves proved that the outerthickness of any planar graph is at most 2. In this paper, we apply the method of deleting spanning disks of embeddings to approximate the thickness and outerthickness of graphs. We first obtain better upper bounds for thickness. We then use a similar approach to provide upper bounds for outerthickness of graphs in terms of…
In this article, we investigate short topological decompositions of non-orientable surfaces and provide algorithms to compute them. Our main result is a polynomial-time algorithm that for any graph embedded in a non-orientable surface computes a canonical non-orientable system of loops so that any loop from the canonical system intersects any edge of the graph in at most 30 points. The existence of such short canonical systems of loops was well known in the orientable case and an open problem in the non-orientable case. Our proof techniques combine recent work of Schaefer-Štefankovič with idea…
Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We prove that the component structure of a graph chosen uniformly at random from the class $\mathcal{S}_g(n,m)$ of all graphs on vertex set $[n]=\{1,\dotsc,n\}$ with $m$ edges embeddable on $\mathbb{S}_g$ features two phase transitions. The first phase transition mirrors the classical phase transition in the Erdős--Rényi random graph $G(n,m)$ chosen uniformly at random from all graphs with vertex set $[n]$ and $m$ edges. It takes place at $m=\frac{n}{2}+O(n^{2/3})$, when a unique largest component, the so-called \emph{giant component},…
It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}γ^n n!$ where $c^{(g)}>0$, and $γ\approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-…
The bondage number b(G) of a graph G is the smallest number of edges of G whose removal from G results in a graph having the domination number larger than that of G. We show that, for a graph G having the maximum vertex degree $Δ(G)$ and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, $b(G)\le \min\{Δ(G)+h+2, Δ(G)+k+1\}$. This generalizes known upper bounds for planar and toroidal graphs.
The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest pat…
The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of the graph, and show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus. Also, we provide stronger upper bounds for graphs with no triangles and graphs with the number of vertices la…
A graph $G$ is equimatchable if any matching in $G$ is a subset of a maximum-size matching. It is known that any $2$-connected equimatchable graph is either bipartite or factor-critical. We prove that for any vertex $v$ of a $2$-connected factor-critical equimatchable graph $G$ and a minimal matching $M$ that isolates $v$ the graph $G\setminus(M\cup\{ v\})$ is either $K_{2n}$ or $K_{n,n}$ for some $n$. We use this result to improve the upper bounds on the maximum size of $2$-connected equimatchable factor-critical graphs embeddable in the orientable surface of genus $g$ to $4\sqrt g+17$ if $g\…
In this paper, the quadrangular genus (4-genus) of the complete graph $K_p$ is shown to be $γ_4 (K_p) = \lceil {p(p-5)}/{8} \rceil +1$ for orientable surfaces. This means that $K_p$ is minimally embeddable in the closed orientable surface of genus $γ_4 (K_p)$ under the constraint that each face has length at least 4. In the most general setting, the genus of the complete graph was established by Ringel and Youngs and was mainly concerned with triangulations of surfaces. Nonetheless, since then a great deal of interest has also been generated in quadrangulations of surfaces. Hartsfield and Ring…
The Bandwidth Theorem of Böttcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a-arrangeable graphs H with Δ(H)<sqrt(n)/log(n), where n is the number of vertices of H. Our result implies that sufficiently large n-vertex graphs G with minimum degree at least (3/4+γ)n contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, Brightwell and Skokan [Combinatorica, to appear] we can also apply ou…
The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Recently Gagarin and Zverovich showed that, for a graph $G$ with maximum degree $Δ(G)$ and embeddable on an orientable surface of genus $h$ and a non-orientable surface of genus $k$, $b(G)\leq\min\{Δ(G)+h+2,Δ+k+1\}$. They also gave examples showing that adjustments of their proofs implicitly provide better results for larger values of $h$ and $k$. In this paper we establish an improved explicit upper bound for $b(G)$, using the Euler characteristic $…
Geometrization of Graphs: Towards Bounding the Chromatic Number via High-Dimensional Embedding
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；CAPT, LMAM and School of Mathematical Sciences, Peking University（北京大学数学科学学院）
Hartsfield and Ringel constructed orientable quadrangular embeddings of the complete graph $K_n$ for $n\equiv 5 \pmod 8$, and nonorientable ones for $n \ge 9$ and $n\equiv 1 \pmod 4$. These provide minimal quadrangulations of their underlying surfaces. We extend these results to determine, for every complete graph $K_n$, $n \ge 4$, the minimum genus, both orientable and nonorientable, for the surface in which $K_n$ has an embedding with all faces of degree at least $4$, and also for the surface in which $K_n$ has an embedding with all faces of even degree. These last embeddings provide sharpne…
We follow the example of Tutte in his construction of the dichromate of a graph (that is, the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, Bollobás-Riordan polynomial and Kruskhal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking v…
The cochromatic number $Z(G)$ of a graph $G$ is the fewest number of colors needed to color the vertices of $G$ so that each color class is a clique or an independent set. In a fractional cocoloring of $G$ a non-negative weight is assigned to each clique and independent set so that for each vertex $v$, the sum of the weights of all cliques and independent sets containing $v$ is at least one. The smallest total weight of such a fractional cocoloring of $G$ is the fractional cochromatic number $Z_f(G)$. In this paper we prove results for the fractional cochromatic number $Z_f(G)$ that parallel r…
Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the interior along surfaces we can adapt the degrees along the boundary of that surface. More efficient is a self-cobordism of G with itself with a host graph discretizing the product of G with an interval. It follows from the fact that Euler curvature is zero everywhere for three dime…
