Coloring graphs by translates in the circle

Polychromatic Colorings on the Hypercube

Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a structured class of colorings, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embed…

具有线性邻域复杂度的奇 coloring 图

Odd coloring graphs with linear neighborhood complexity

该研究证明了具有线性邻域复杂度的图类具有有界不良奇色数，并展示了圆图、双宽度有界类等情形下的结论。

First non-trivial upper bound on the circular chromatic number of the plane

We consider circular version of the famous Nelson-Hadwiger problem. It is know that 4 colors are necessary and 7 colors suffice to color the euclidean plane in such a way that points at distance one get different colors. In $r$-circular coloring we assign arcs of length one of a circle with a perimeter $r$ in such a way that points at distance one get disjoint arcs. In this paper we show the existence of $r$-circular coloring for $r=4+\frac{4\sqrt{3}}{3}\approx 6.30$. It is the first result with $r$-circular coloring of the plane with $r$ smaller than 7. We also show $r$-circular coloring of t…

Collective dynamics of phase-repulsive oscillators solves graph coloring problem

We show how to couple phase-oscillators on a graph so that collective dynamics "searches" for the coloring of that graph as it relaxes toward the dynamical equilibrium. This translates a combinatorial optimization problem (graph coloring) into a functional optimization problem (finding and evaluating the global minimum of dynamical non-equilibrium potential, done by the natural system's evolution). Using a sample of graphs, we show that our method can serve as a viable alternative to the traditional combinatorial algorithms. Moreover, we show that, with the same computational cost, our method …

Chip games and paintability

We prove that the difference between the paint number and the choice number of a complete bipartite graph $K_{N,N}$ is $Θ(\log \log N )$. That answers the question of Zhu (2009) whether this difference, for all graphs, can be bounded by a common constant. By a classical correspondence, our result translates to the framework of on-line coloring of uniform hypergraphs. This way we obtain that for every on-line two coloring algorithm there exists a k-uniform hypergraph with $Θ(2^k )$ edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games.

Geometry based heuristics for unit disk graphs

Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of th…

On Coloring the Arcs of Biregular Graphs

Recalling each edge of a graph $H$ has 2 oppositely oriented arcs, each vertex $v$ of $H$ is identified with the set of arcs, denoted $(v,e)$, departing from $v$ along the edges $e$ of $H$ incident to $v$. Let $H$ be a $(λ,μ)$-biregular graph with bipartition $(Y,X)$, where $|Y|=kμ$ and $|X|=kλ$, ($0<k,λ,μ\in\mathbb{Z}$). We consider the problem, for each edge $e=yx$ in $H$, of assigning, a color (given by an element) of $Y$, resp. $X$, to the arc $(y,e)$, resp. $(x,e)$, so that each color is assigned exactly once in the set of arcs departing from each vertex of $H$. Furthermore, we set such a…

Signed bipartite circular cliques and a bipartite analogue of Grötzsch's theorem

The notion of the circular coloring of signed graphs is a recent one that simultaneously extends both notions of the circular coloring of graphs and $0$-free coloring of signed graphs. A circular $r$-coloring of a signed graph $(G, σ)$ is to assign points of a circle of circumference $r$, $r\geq 2$, to the vertices of $G$ such that vertices connected by a positive edge are at circular distance at least $1$ and vertices connected by a negative edge are at circular distance at most $\frac{r}{2}-1$. The infimum of all $r$ for which $(G, σ)$ admits a circular $r$-coloring is said to be the circula…

Circular $(4-ε)$-coloring of some classes of signed graphs

A circular $r$-coloring of a signed graph $(G, σ)$ is an assignment $ϕ$ of points of a circle $C_r$ of circumference $r$ to the vertices of $(G, σ)$ such that for each positive edge $uv$ of $(G, σ)$ the distance of $ϕ(v)$ and $ϕ(v)$ is at least 1 and for each negative edge $uv$ the distance of $ϕ(u)$ from the antipodal of $ϕ(v)$ is at least 1. The circular chromatic number of $(G, σ)$, denoted $χ_c(G, σ)$, is the infimum of $r$ such that $(G, σ)$ admits a circular $r$-coloring. This notion is recently defined by Naserasr, Wang, and Zhu who, among other results, proved that for any signed $d$-d…

Circular chromatic number of signed graphs

A signed graph is a pair $(G, σ)$, where $G$ is a graph and $σ: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, σ)$ a circular $r$-coloring of $(G, σ)$ is an assignment $ψ$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $σ(e)=+$, then $ψ(u)$ and $ψ(v)$ have distance at least $1$, and if $σ(e)=-$, then $ψ(v)$ and the antipodal of $ψ(u)$ have distan…

List-coloring the Square of a Subcubic Graph

The {\em square} $G^2$ of a graph $G$ is the graph with the same vertex set as $G$ and with two vertices adjacent if their distance in $G$ is at most 2. Thomassen showed that every planar graph $G$ with maximum degree $Δ(G)=3$ satisfies $χ(G^2)\leq 7$. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of $G^2$ equals the chromatic number of $G^2$, that is $χ_l(G^2)=χ(G^2)$ for all $G$. If true, this conjecture (together with Thomassen's result) implies that every planar graph $G$ with $Δ(G)=3$ satisfies $χ_l(G^2)\leq 7$. We prove that every connected graph (not …

Coloring Discrete Manifolds

Discrete d-manifolds are classes of finite simple graphs which can triangulate classical manifolds but which are defined entirely within graph theory. We show that the chromatic number X(G) of a discrete d-manifold G is sandwiched between d+1 and 2(d+1). From the general identity X(A+B)=X(A)+X(B) for the join A+B of two finite simple graphs, it follows that there are (2k)-spheres with chromatic number X=(3k+1) and (2k-1)-spheres with chromatic number X=3k. Examples of 2-manifolds with X(G)=5 have been known since the pioneering work of Fisk. Current data support the that the ceiling function o…

Extending Precolorings to Distinguish Group Actions

Given a group $Γ$ acting on a set $X$, a $k$-coloring $ϕ:X\to\{1,\dots,k\}$ of $X$ is distinguishing with respect to $Γ$ if the only $γ\in Γ$ that fixes $ϕ$ is the identity action. The distinguishing number of the action $Γ$, denoted $D_Γ(X)$, is then the smallest positive integer $k$ such that there is a distinguishing $k$-coloring of $X$ with respect to $Γ$. This notion has been studied in a number of settings, but by far the largest body of work has been concerned with finding the distinguishing number of the action of the automorphism group of a graph $G$ upon its vertex set, which is refe…

有限射影空间的色指数

The chromatic index of finite projective spaces

本文将确定有限射影空间 PG$(n,q)$ 色指数的问题转化为检验特定 PG$(3,q)$ 和 PG$(4,q)$ 的存在性，并证明了对于任意奇数 $n$ 及 $q\in\{3,4,8,16\}$ 的色指数及其平行性的存在。

Graph Isomorphism Restricted by Lists

The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical computer science. For graphs $G$ and $H$, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list restricted graph isomorphism (ListIso) is NP-complete: for each $u \in V(G)$, we are given a list ${\mathfrak L}(u) \subseteq V(H)$ of possible images of $u$. After 35 years, we revive the study of this problem and consider which results for GraphIso translate to ListIso. We prove the following: 1) When GraphIso is GI-complete for a class of graphs, it translates i…

Note on the chromatic number of Minkowski planes: the regular polygon case

The famous Hadwiger-Nelson problem asks for the minimum number of colors needed to color the points of the Euclidean plane so that no two points unit distance apart are assigned the same color. In this note we consider a variant of the problem in Minkowski metric planes, where the unit circle is a regular polygon of even and at most 22 vertices. We present a simple lattice-sublattice coloring scheme that uses 6 colors, proving that the chromatic number of the Minkowski planes above are at most 6. This result is new for regular polygons having more than 8 vertices.

Distinguishing extension numbers for $\mathbf R^n$ and $S^n$

In the setting of a group $Γ$ acting faithfully on a set $X$, a $k$-coloring $c: X\rightarrow \{1, 2, ..., k\}$ is called $Γ$-distinguishing if the only element of $Γ$ that fixes $c$ is the identity element. The distinguishing number $D_Γ(X)$ is the minimum value of $k$ such that a $Γ$-distinguishing $k$-coloring of $X$ exists. Now, fixing $k= D_Γ(X)$, a subset $W\subset X$ with trivial pointwise stabilizer satisfies the precoloring extension property $P(W)$ if every precoloring $c: X-W\rightarrow \{1, ..., k\}$ can be extended to a $Γ$-distinguishing $k$-coloring of $X$. The distinguishing ex…

Triangle-free geometric intersection graphs with large chromatic number

Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$…

关于 $Δ=3$ 的符号图的 Brooks 定理

Brooks' theorem for signed graphs with $Δ=3$

本文研究最大度为3的符号图的圆形色数上界，证明除 $(K_4,-)$ 分量外均满足 $χ_c\leq 10/3$，并通过 Petersen 图证明该界紧。

Location of zeros for the partition function of the Ising model on bounded degree graphs

The seminal Lee-Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in $\mathbb C$. In fact the union of the zeros of all graphs is dense on the unit circle. In this paper we study the location of the zeros for the class of graphs of bounded maximum degree $d\geq 3$, both in the ferromagnetic and the anti-ferromagnetic case. We determine the location exactly as a function of the inverse temperature and the degree $d$. An important step in our approach is to translate to the setting of complex dynamics and analyze a …

关于正冲突自由和奇色数的有界性

Boundedness for proper conflict-free and odd colorings

本文改进了无爪图的正冲突自由色数界，并展示了凸轮图和排列图的线性有界性。