We consider the $t$-improper chromatic number of the Erd{\H o}s-R{é}nyi random graph $G(n,p)$. The t-improper chromatic number $χ^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$, then this is the usual notion of proper colouring. When the edge probability $p$ is constant, we provide a detailed description of the asymptotic behaviour of $χ^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.
Let $G$ be a graph whose each component has order at least 3. Let $s : E(G) \rightarrow \mathbb{Z}_k$ for some integer $k\geq 2$ be an improper edge coloring of $G$ (where adjacent edges may be assigned the same color). If the induced vertex coloring $c : V (G) \rightarrow \mathbb{Z}_k$ defined by $c(v) = \sum_{e\in E_v} s(e) \mbox{ in } \mathbb{Z}_k,$ (where the indicated sum is computed in $\mathbb{Z}_k$ and $E_v$ denotes the set of all edges incident to $v$) results in a proper vertex coloring of $G$, then we refer to such a coloring as an improper twin $k$-edge coloring. The minimum $k$ fo…
In this paper we study the set chromatic number of a random graph $G(n,p)$ for a wide range of $p=p(n)$. We show that the set chromatic number, as a function of $p$, forms an intriguing zigzag shape.
A random geometric graph $G_n$ is given by picking $n$ vertices in $\mathbb{R}^d$ independently under a common bounded probability distribution, with two vertices adjacent if and only if their $l^p$-distance is at most $r_n$. We investigate the distant-$l$ chromatic number $χ_l(G_n)$ of $G_n$ for $l\ge1$. Complete picture of the ratios of $χ_l(G_n)$ to the chromatic number $χ(G_n)$ are given in the sense of almost sure convergence.
Let G(n,d) be the random d-regular graph on n vertices. For any integer k exceeding a certain constant k_0 we identify a number d_{k-col} such that G(n,d) is k-colorable w.h.p. if d<d_{k-col} and non-k-colorable w.h.p. if d>d_{k-col}.
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 resili…
A graph is \textit{locally irregular} if the neighbors of every vertex $v$ have degrees distinct from the degree of $v$. \textit{locally irregular edge-coloring} of a graph $G$ is an (improper) edge-coloring such that the graph induced on the edges of any color class is locally irregular. It is conjectured that $3$ colors suffice for a locally irregular edge-coloring. Recently, Bensmail et al. (Bensmail, Merker, Thomassen: Decomposing graphs into a constant number of locally irregular subgraphs, {\em European J. Combin.}, 60:124--134, 2017) settled the first constant upper bound for the proble…
Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each V_i. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by adding k \lceil n/k \rceil - n isolated vertices is strongly k-colorable. The strong chromatic number of G is the minimum k for which G is strongly k-colorable. In this paper, we study the behavior…
A clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study the asymptotic behaviour of the clique chromatic number of the random graph G(n,p) for a wide range of edge-probabilities p=p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function.
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 $n \rightarrow \infty$.
Given any integer d >= 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k then the chromatic number is either k+1 or k+2.
Let 0<p<1 be fixed. Shamir and Spencer proved in the 1980s that the chromatic number of a random graph in G(n,p) is concentrated in an interval of length about n^{1/2}. In this explanatory note, we give a proof of a result due due Noga Alon, showing that the chromatic number is concentrated in an interval of length about n^{1/2}/log n.
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.
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. In this paper we analyze the asymptotic behavior of this parameter for a random graph G_{n,p}. We show that with high probability the game chromatic number of G_{n,p} is at least twice its chromatic number but, up to a multiplicative constant, has the sa…
Suppose that two players take turns coloring the vertices of a given graph G with k colors. In each move the current player colors a vertex such that neighboring vertices get different colors. The first player wins this game if and only if at the end, all the vertices are colored. The game chromatic number $χ_g(G)$ is defined as the smallest k for which the first player has a winning strategy. Recently, Bohman, Frieze and Sudakov [Random Structures and Algorithms 2008] analysed the game chromatic number of random graphs and obtained lower and upper bounds of the same order of magnitude. In thi…
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 …
Given independent random points $X_1,...,X_n\in\eR^d$ with common probability distribution $ν$, and a positive distance $r=r(n)>0$, we construct a random geometric graph $G_n$ with vertex set $\{1,...,n\}$ where distinct $i$ and $j$ are adjacent when $\norm{X_i-X_j}\leq r$. Here $\norm{.}$ may be any norm on $\eR^d$, and $ν$ may be any probability distribution on $\eR^d$ with a bounded density function. We consider the chromatic number $χ(G_n)$ of $G_n$ and its relation to the clique number $ω(G_n)$ as $n \to \infty$. Both McDiarmid and Penrose considered the range of $r$ when $r \ll (\frac{\l…
The \textit{Distinguishing Chromatic Number} of a graph $G$, denoted $χ_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $ϕ$ of the graph $G$ fixes each color class of $G$. In this paper, we consider random Cayley graphs $Γ(A,S)$ defined over certain abelian groups $A$ and show that with probability at least $1-n^{-Ω(\log n)}$ we have, $χ_D(Γ)\leχ(Γ) + 1$.
In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model $\Gnd$ for $d=o(n^{1/5})$ is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that of the binomial random graph model $\Gnp$ with $p=\frac{d}{n}$. Our proof is largely based on ideas of Alon and Krivelevich who proved this two-point concentration result for $\Gnp$ for $p=n^{-δ}$ where $δ>1/2$. The main tool used to derive such a result is a careful analysis of the distribution…
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.
Recently, Kupavskii~[{\it On random subgraphs of {K}neser and {S}chrijver graphs. J. Combin. Theory Ser. A, {\rm 2016}.}] investigated the chromatic number of random Kneser graphs $\KG_{n,k}(ρ)$ and proved that, in many cases, the chromatic numbers of the random Kneser graph $\KG_{n,k}(ρ)$ and the Kneser graph $\KG_{n,k}$ are almost surely closed. He also marked the studying of the chromatic number of random Kneser hypergraphs $\KG^r_{n,k}(ρ)$ as a very interesting problem. With the help of $\Z_p$-Tucker lemma, a combinatorial generalization of the Borsuk-Ulam theorem, we generalize Kupavskii'…
In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k-1 or k. If moreover d>(2k-3)log(k-1), then the value k-1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k+1. Our proof applies the small…
Let G be an abelian group of cardinality N, where (N,6) = 1, and let A be a random subset of G. Form a graph Gamma_A on vertex set G by joining x to y if and only if x + y is in A. Then, almost surely as N tends to infinity, the chromatic number chi(Gamma_A) is at most (1 + o(1))N/2 log_2 N. This is asymptotically sharp when G = Z/NZ, N prime. Presented at the conference in honour of Bela Bollobas on his 70th birthday, Cambridge August 2013.
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.
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…
We consider the problem of $k$-colouring a random $r$-uniform hypergraph with $n$ vertices and $cn$ edges, where $k$, $r$, $c$ remain constant as $n$ tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case $r=2$, must have one of two easily computable values as $n$ tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs.
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…
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…
A proper 2-tone $k$-coloring of a graph is a labeling of the vertices with elements from $\binom{[k]}{2}$ such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number of a graph $G$, denoted $τ_2(G)$ is the smallest $k$ such that $G$ admits a proper 2-tone $k$ coloring. In this paper, we prove that w.h.p. for $p\ge Cn^{-1/4}\ln^{9/4}n$, $τ_2(G_{n,p})=(2+o(1))χ(G_{n,p})$ where $χ$ represents the ordinary chromatic number. For sparse random graphs with $p=c/n$, $c$ constant, we prove that $τ_2(G_{n,p}) = \lceil{{\sqrt{8Δ+1…
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…
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…
The chromatic threshold $δ_χ(H,p)$ of a graph $H$ with respect to the random graph $G(n,p)$ is the infimum over $d > 0$ such that the following holds with high probability: the family of $H$-free graphs $G \subset G(n,p)$ with minimum degree $δ(G) \ge dpn$ has bounded chromatic number. The study of $δ_χ(H) :=δ_χ(H,1)$ was initiated in 1973 by Erdős and Simonovits. Recently $δ_χ(H)$ was determined for all graphs $H$. It is known that $δ_χ(H,p) =δ_χ(H)$ for all fixed $p \in (0,1)$, but that typically $δ_χ(H,p) \ne δ_χ(H)$ if $p = o(1)$. Here we study the problem for sparse random graphs. We dete…
Amit, Linial, and Matou\vsek (Random lifts of graphs III: independence and chromatic number, Random Struct. Algorithms, 2001) have raised the following question: Is the chromatic number of random $h$-lifts of $K_5$ asymptotically (for $h\to\infty$) almost surely equal to a single number? In this paper, we offer the following partial result: The chromatic number of a random lift of $K_5\setminus e$ is asymptotically almost surely three.
Green showed that there exist constants $C_1,C_2>0$ such that the clique number $ω$ of the random Cayley graph on $\mathbb{F}_2^n$ satisfies $\lim_{n\to\infty}\mathbb{P}(C_1n\log n < ω< C_2n\log n)=1$. In this paper we find the best possible $C_1$ and $C_2$. Moreover, we prove that for $n$ in a set of density $1$, clique number is actually concentrated on a single value. As a simple consequence of these results, we also prove the one-point concentration result for the chromatic number, thus proving the $\mathbb{F}_2^n$ analogue of the famous conjecture by Bollobás and giving almost the complet…
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…
The chromatic threshold $δ_χ(H,p)$ of a graph $H$ with respect to the random graph $G(n,p)$ is the infimum over $d > 0$ such that the following holds with high probability: the family of $H$-free graphs $G \subset G(n,p)$ with minimum degree $δ(G) \ge dpn$ has bounded chromatic number. The study of the parameter $δ_χ(H) := δ_χ(H,1)$ was initiated in 1973 by Erdős and Simonovits, and was recently determined for all graphs $H$. In this paper we show that $δ_χ(H,p) = δ_χ(H)$ for all fixed $p \in (0,1)$, but that typically $δ_χ(H,p) \ne δ_χ(H)$ if $p = o(1)$. We also make significant progress towa…
Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression…
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge dens…
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…
We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős-Rényi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3.
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 …
