Chromatic Number Of Random Graphs Research Articles

On the Concentration of the Chromatic Number of Random Graphs

Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph $G_{n,p}$ is concentrated in an interval of length at most $\omega\sqrt{n}$, and in the 1990s Alon showed that an interval of length $\omega\sqrt{n}/\log n$ suffices for constant edge-probabilities $p\in (0,1)$. We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case ${p=p(n) \to 0}$, and uncover a surprising concentration `jump' of the chromatic number in the very dense case ${p=p(n) \to 1}$.

The jump of the clique chromatic number of random graphs

AbstractThe clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Prałat noted that around the clique chromatic number of the random graph changes by when we increase the edge‐probability by , but left the details of this surprising “jump” phenomenon as an open problem. We settle this problem, that is, resolve the nature of this polynomial “jump” of the clique chromatic number of the random graph around edge‐probability . Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of up to logarithmic factors for any edge‐probability .

Lower Bounds on the Chromatic Number of Random Graphs

We prove that a formula predicted on the basis of non-rigorous physics arguments [Zdeborová 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, we show how asymptotic formulas for large degrees that were previously obtained by lengthy and complicated combinatorial arguments can be re-derived easily from these new results.

The eternal game chromatic number of random graphs

The eternal vertex colouring game, recently introduced byKlostermeyer and Mendoza (2018), is a version of the vertex colouring game, a well studied graph game. The game is played by two players, Alice and Bob on a graph G with a set of colours {1,…,k}. The game consists of rounds, such that in each round, every vertex is coloured exactly once. In the first round, players alternate turns with Alice playing first. During their turn, each player first picks yet uncoloured vertex and then colours it by any colour so that the resulting partial colouring of the graph is proper (if at least one such colour exists). During all further rounds, players keep choosing vertices alternately. After choosing a vertex, the player assigns a colour to the vertex which is distinct from its current colour such that the resulting colouring is still proper. Each vertex retains its colour between rounds until it is recoloured. Bob wins if at any point the chosen vertex does not have a legal recolouring, while Alice wins if the game is continued indefinitely. The eternal game chromatic number χg∞(G) is the smallest number k such that Alice has a winning strategy.In this note, we study the eternal game chromatic number of random graphs. We show that with high probability χg∞(Gn,p)=(p2+o(1))n for odd n, and also for even n when p=1k for some k∈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 by Klostermeyer and Mendoza.

The set chromatic number of random graphs

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.

The Chromatic Number of Random Graphs for Most Average Degrees

For a fixed number d> 0 and n large, let G(n,d/n) be the random graph on n vertices in which any two vertices are connected with probability d/n independently. The problem of determining the chromatic number of G(n,d/n) goes back to the famous 1960 article of Erdős and Renyi that started the theory of random graphs [Magayar Tud. Akad. Mat. Kutato Int. Kozl. 5 (1960) 17–61]. Progress culminated in the landmark paper of Achlioptas and Naor [Ann. Math. 162 (2005) 1333–1349], in which they calculate the chromatic number precisely for all d in a set S⊂ (0,∞) of asymptotic density limz→∞ z ∫z 0 1S = 2 , and up to an additive error of one for the remaining d. Here we obtain a near-complete answer by determining the chromatic number of G(n,d/n) for all d in a set of asymptotic density 1.

The Game Chromatic Number of Dense Random Graphs

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 vertices are colored. The game chromatic number χg(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 this paper we improve existing results and show that with high probability, the game chromatic number χg(Gn,p) of dense random graphs with p ≥ e-o(log n) is asymptotically twice as large as the ordinary chromatic number χ(Gn,p).

Indicated coloring of graphs

We study a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben’s strategy) is called the indicated chromatic number of G, and denoted by χi(G). We approach the question how much χi(G) differs from the usual chromatic number χ(G). In particular, whether there is a function f such that χi(G)⩽f(χ(G)) for every graph G. We prove that f cannot be linear with leading coefficient less than 4/3. On the other hand, we show that the indicated chromatic number of random graphs is bounded roughly by 4χ(G). We also exhibit several classes of graphs for which χi(G)=χ(G) and show that this equality for any class of perfect graphs implies Clique-Pair Conjecture for this class of graphs.

On the Strong Chromatic Number of Random Graphs

Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colourable if, for every partition of V(G) into disjoint sets V1 ∪ ··· ∪ Vr, all of size exactly k, there exists a proper vertex k-colouring of G with each colour appearing exactly once in each Vi. In the case when k does not divide n, G is defined to be strongly k-colourable if the graph obtained by adding $k \big\lceil \frac{n}{k} \big\rceil - n$ isolated vertices is strongly k-colourable. The strong chromatic number of G is the minimum k for which G is strongly k-colourable. In this paper, we study the behaviour of this parameter for the random graph Gn,p. In the dense case when p ≫ n−1/3, we prove that the strong chromatic number is a.s. concentrated on one value Δ + 1, where Δ is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.

On the chromatic number of random graphs

In this paper we consider the classical Erdős–Rényi model of random graphs G n , p . We show that for p = p ( n ) ⩽ n − 3 / 4 − δ , for any fixed δ > 0 , the chromatic number χ ( G n , p ) is a.a.s. ℓ, ℓ + 1 , or ℓ + 2 , where ℓ is the maximum integer satisfying 2 ( ℓ − 1 ) log ( ℓ − 1 ) ⩽ p ( n − 1 ) .

The game chromatic number of random graphs

AbstractGiven 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 χg(G) is the minimum k for which the first player has a winning strategy. In this study, we analyze the asymptotic behavior of this parameter for a random graph Gn,p. We show that with high probability, the game chromatic number of Gn,p is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008

On the Chromatic Number of Random Graphs with a Fixed Degree Sequence

Let d=1≤d1≤ d2≤···.≤ dn be a non-decreasing sequence of n positive integers, whose sum is even. Let denote the set of graphs with vertex set [n]={1,2,. . .., n} in which the degree of vertex i is di. Let Gn,d be chosen uniformly at random from . Let d=(d1+d2+···.+dn)/n be the average degree. We give a condition on d under which we can show that w.h.p. the chromatic number of is Θ(d/ln d). This condition is satisfied by graphs with exponential tails as well those with power law tails.

An improved algorithm for approximating the chromatic number of [formula omitted

Answering a question of Krivelevich and Vu [M. Krivelevich, V.H. Vu, Approximating the independence number and the chromatic number in expected polynomial time, J. Combin. Optimization 6 (2002) 143–155], we present an algorithm for approximating the chromatic number of random graphs G n , p within a factor of O ( n p / ln ( n p ) ) in polynomial expected time. The algorithm applies to edge probabilities c 0 / n ⩽ p ⩽ 0.99 , where c 0 > 0 is a certain constant.

MAX k‐CUT and approximating the chromatic number of random graphs

AbstractWe consider the MAX k‐CUT problem on random graphs Gn,p. First, we bound the probable weight of a MAX k‐CUT using probabilistic counting arguments and by analyzing a simple greedy heuristic. Then, we give an algorithm that approximates MAX k‐CUT in expected polynomial time, with approximation ratio 1 + O((np)‐1/2). Our main technical tool is a new bound on the probable value of Frieze and Jerrum's semidefinite programming (SDP)‐relaxation of MAX k‐CUT on random graphs. To obtain this bound, we show that the value of the SDP is tightly concentrated. As a further application of our bound on the probable value of the SDP, we obtain an algorithm for approximating the chromatic number of Gn,p, 1/n ≤ p ≤ 0.99, within a factor of O((np)1/2) in polynomial expected time, thereby answering a question of Krivelevich and Vu. We give similar algorithms for random regular graphs. The techniques for studying the SDP apply to a variety of SDP relaxations of further NP‐hard problems on random structures and may therefore be of independent interest. For instance, to bound the SDP we estimate the eigenvalues of random graphs with given degree sequences. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

The concentration of the chromatic number of random graphs

We prove that for every constant δ>0 the chromatic number of the random graphG(n, p) withp=n−1/2−δ is asymptotically almost surely concentrated in two consecutive values. This implies that for any β<1/2 and any integer valued functionr(n)≤O(nβ) there exists a functionp(n) such that the chromatic number ofG(n,p(n)) is preciselyr(n) asymptotically almost surely.

Generalized Chromatic Numbers of Random Graphs

Given a hereditary (closed under taking induced subgraphs) class of graphs $\mathcal{P}$, the $\mathcal{P}$-chromatic number of a graph G, denoted $\chi _\mathcal{P} ( G )$, is defined to be the least integer k such that $V( G )$ admits a partition into k subsets each of which induce a member of $\mathcal{P}$. The $\mathcal{P}$-chromatic number of random graphs G on n vertices with fixed edge probability $0 < p < 1$ is studied and it is shown that $\chi _\mathcal{P} ( G ) \sim cn$ in case $\mathcal{P} < \infty $ and $\chi _\mathcal{P} ( G ) = \Theta ( n/\log n )$ in case $| \mathcal{P} | = \infty $. Also considered are generalized edge chromatic numbers.

A note on the sharp concentration of the chromatic number of random graphs

A note on the sharp concentration of the chromatic number of random graphs

The chromatic number of random graphs

Let χ(G(n, p)) denote the chromatic number of the random graphG(n, p). We prove that there exists a constantd 0 such that fornp(n)>d 0,p(n)→0, the probability that $$\frac{{np}}{{2 log np}}\left( {1 + \frac{{\log log np - 1}}{{\log np}}} \right)< \chi (G(n,p))< \frac{{np}}{{2 log np}}\left( {1 + \frac{{30 \log \log np}}{{\log np}}} \right)$$ tends to 1 asn→∞.

On the chromatic number of random graphs

AbstractWe consider random graphs Gn,p with fixed edge‐probability p. We refine an argument of Bollobás to show that almost all such graphs have chromatic number equal to n/{2 logb n − 2 logb logb n + O(1)} where b = 1/(1 − p).

The chromatic number of random graphs at the double-jump threshold

A crucial step in the Erdos-Renyi (1960) proof that the double-jump threshold is also the planarity threshold for random graphs is shown to be invalid. We prove that whenp=1/n, almost all graphs do not contain a cycle with a diagonal edge, contradicting Theorem 8a of Erdos and Renyi (1960). As a consequence, it is proved that the chromatic number is 3 for almost all graphs whenp=1/n.