What is the minimum number of edges that have to be added to the random graph $G=G_{n,0.5}$ in order to increase its chromatic number $χ=χ(G)$ by one percent ? One possibility is to add all missing edges on a set of $1.01 χ$ vertices, thus creating a clique of chromatic number $1.01 χ$. This requires, with high probability, the addition of $Ω(n^2/\log^2 n)$ edges. We show that this is tight up to a constant factor, consider the question for more general random graphs $G_{n,p}$ with $p=p(n)$, and study a local version of the question as well.
The question is motivated by the study of the resilience of graph properties, initiated by the second author and Vu, and improves one of their results.
It is well known that a random subgraph of the complete graph $K_n$ has chromatic number $Θ(n/\log n)$ w.h.p. Boris Bukh asked whether the same holds for a random subgraph of any $n$-chromatic graph, at least in expectation. In this paper it is shown that for every graph, whose fractional chromatic number is at least $n$, the fractional chromatic number of its random subgraph is at least $n/(8\log_2(4n))$ with probability more than $1-\frac{1}{2n}$. This gives the affirmative answer for a strengthening of Bukh's question for the fractional chromatic number.
We estimate the likely values of the chromatic and independence numbers of the random $r$-uniform $d$-regular hypergraph on $n$ vertices for fixed $r$, large fixed $d$, and $n \rightarrow \infty$.
We show that the chromatic number of $G_{n, \frac 12}$ is not concentrated on fewer than $n^{\frac 14 - \varepsilon}$ consecutive values. This addresses a longstanding question raised by Erdős and several other authors.
An $n$-lift of a graph $G$ is a graph from which there is an $n$-to-$1$ covering map onto $G$. Amit, Linial, and Matou\v sek (2002) raised the question of whether the chromatic number of a random $n$-lift of $K_5$ is concentrated on a single value. We consider this problem for $G=K_{d+1}$, and show that for fixed $d\ge 3$ the chromatic number of a random lift of $K_d$ is (asymptotically almost surely) either $k$ or $k+1$, where $k$ is the smallest integer satisfying $d < 2k \log k$. Moreover, we show that, for roughly half of the values of $d$, the chromatic number is concentrated on $k$. The …
We consider the following game, played on a $k$-uniform hypergraph $H$. There are $q$ colors available and two players take it in turns to color vertices. A partial coloring is proper if no edge is mono-chromatic. One player, A, wishes to color all the vertices and the other player, B, wishes to prevent this. The {\em game chromatic number} $χ_g(H)$ is the minimum number of colors for which A has a winning strategy. We consider this in the context of a random $k$-uniform hypergraph and prove upper and lower bounds that hold w.h.p.
The $\!{}\bmod k$ chromatic index of a graph $G$ is the minimum number of colors needed to color the edges of $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1\!\!\pmod k$. Recently, the authors proved that the $\!{}\bmod k$ chromatic index of every graph is at most $198k-101$, improving, for large $k$, a result of Scott [Discrete Math. 175, 1-3 (1997), 289-291]. Here we study the $\!{}\bmod k$ chromatic index of random graphs. We prove that for every integer $k\geq2$, there is $C_k>0$ such that if $p\geq C_kn^{-1}\log{n}$ and $n(1-p) \rightarrow…
Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of $G$ are colored. The game chromatic number χ_g(G) is the minimum k for which the first player has a winning strategy. The paper \cite{BFS} began the analysis of the asymptotic behavior of this parameter for a random graph G_{n,p}. This paper provides some further analysis for graphs with constant average degree i.e. np=O(1) and for random regular graphs.
We prove that a formula predicted on the basis of non-rigorous physics arguments [Zdeborova and Krzakala: Phys. Rev. E (2007)] provides a lower bound on the chromatic number of sparse random graphs. The proof is based on the interpolation method from mathematical physics. In the case of random regular graphs the lower bound can be expressed algebraically, while in the case of the binomial random we obtain a variational formula. As an application we calculate improved explicit lower bounds on the chromatic number of random graphs for small (average) degrees. Additionally, show how asymptotic fo…
The clique chromatic number of a graph is the minimum number of colors required to assign to its vertex set so that no inclusion maximal clique is monochromatic. McDiarmid, Mitsche and Prałat proved that the clique chromatic number of the binomial random graph $G\left(n,\frac{1}{2}\right) $ is at most $\left(\frac{1}{2}+o(1)\right)\log_2n$ with high probability. Alon and Krivelevich showed that it is greater than $\frac{1}{2000}\log_2n$ with high probability and suggested that the right constant in front of the logarithm is $\frac{1}{2}.$ We prove their conjecture and, beyond that, obtain a ti…
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…
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…
The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this note, we study the eternal game chromatic number of random graphs. We show that with high probability $χ_{g}^{\infty}(G_{n,p}) = (\frac{p}{2} + o(1))n$ for odd $n$, and also for even $n$ when $p=\frac{1}{k}$ for some $k \in \mathbb{N}$. The upper bound applies for even $n$ and any other value of $p$ as well, but we conjecture in this case this upper bound is not sharp. Finally, we answer a question posed…
The acyclic chromatic index of a graph $G$ is the least number of colors needed to properly color its edges so that none of its cycles is bichromatic. In this work, we show that $2Δ-1$ colors are sufficient to produce such a coloring, where $Δ$ is the maximum degree of the graph. In contrast with most extant randomized algorithmic approaches to the chromatic index, where the algorithms presuppose enough colors to guarantee properness deterministically and use randomness only to deal with the bichromatic cycles, our randomized, Moser-type algorithm produces a not necessarily proper random color…
An equitable colouring of a graph $G$ is a colouring of the vertices of $G$ so that no two adjacent vertices are coloured the same and, additionally, the colour class sizes differ by at most $1$. The equitable chromatic number $χ_=(G)$ is the minimum number of colours required for this. We study the equitable chromatic number of the dense random graph $G(n,m)$, where $m = \left\lfloor p {n \choose 2} \right \rfloor $ and $0<p< 0.86$ is constant. It is a well-known question of Bollobás whether for $p=1/2$ there is a function $f(n) \rightarrow \infty$ so that for any sequence of intervals of len…
The chromatic number $χ(G)$ of a graph $G$ is defined as the minimum number of colours required for a vertex colouring where no two adjacent vertices are coloured the same. The chromatic number of the dense random graph $G \sim G(n,p)$ where $p \in (0,1)$ is constant has been intensively studied since the 1970s, and a landmark result by Bollobás in 1987 first established the asymptotic value of $χ(G)$. Despite several improvements of this result, the exact value of $χ(G)$ remains open. In this paper, new upper and lower bounds for $χ(G)$ are established. These bounds are the first ones that ma…
色多项式和相关的图不变量是图论和统计物理学的中心对象。然而，计算困难，到目前为止，限制了研究这些多项式的图形，要么非常小，非常稀疏或高度结构化。最近的算法进展（Timme等人，2009 New J.Phys.11023001）现在使得可以计算任意结构和边数的中等大小的图的色多项式。在这里，我们提出了色多项式的合奏随机图与多达30个顶点，在整个范围内的边缘密度。我们特别关注复平面中多项式的零点的位置。结果表明，随机图的色零点具有非常一致的布局。特别地，交叉点，具有非零虚部的色零点接近实轴的点，在大部分密度范围内与平均度线性地缩放。虽然所获得的标度律是纯经验的，如果他们继续持有一般有重大的影响：交叉点的色零在热力学极限分离系统与零基态熵的系统与积极的基态熵，后者是热力学第三定律的例外。
We prove that in sparse Erdős-Rényi graphs of average degree $d$, the vector chromatic number (the relaxation of chromatic number coming from the Lovàsz theta function) is typically $\tfrac{1}{2}\sqrt{d} + o_d(1)$. This fits with a long-standing conjecture that various refutation and hypothesis-testing problems concerning $k$-colorings of sparse Erdős-Rényi graphs become computationally intractable below the `Kesten-Stigum threshold' $d_{KS,k} = (k-1)^2$. Along the way, we use the celebrated Ihara-Bass identity and a carefully constructed non-backtracking random walk to prove two deterministic…
Given a graph $G$ and $p \in [0,1]$, let $G_p$ denote the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. Alon, Krivelevich, and Sudokov proved $\mathbb{E} [χ(G_p)] \geq C_p \frac{χ(G)}{\log |V(G)|}$, and Bukh conjectured an improvement of $\mathbb{E}[χ(G_p)] \geq C_p \frac{χ(G)}{\log χ(G)}$. We prove a new spectral lower bound on $\mathbb{E}[χ(G_p)]$, as progress towards Bukh's conjecture. We also propose the stronger conjecture that for any fixed $p \leq 1/2$, among all graphs of fixed chromatic number, $\mathbb{E}[χ(G_p)]$ is minimized by the complet…
The chromatic number $χ(G)$ of a graph $G$, that is, the smallest number of colors required to color the vertices of $G$ so that no two adjacent vertices are assigned the same color, is a classic and extensively studied parameter. Here we consider the case where $G$ is a random block graph, also known as the stochastic block model. The vertex set is partitioned into $k\in\mathbb{N}$ parts $V_1, \dotsc, V_k$, and for each $1 \le i\le j\le k$, two vertices $u \in V_i, v\in V_j$ are connected by an edge with some probability $p_{ij} \in (0,1)$ independently. Our main result pins down the typical …
