On the chromatic number of random graphs

We consider random graphs G_n,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 log_b n ? 2 log_b log_b n + O(1)} where b = 1/(1 ? p).     

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 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→∞.     

The chromatic number of random graphs

For a fixed probabilityp, 0<p<1, almost every random graphG _n,p has chromatic number $$\left( {\frac{1}{2} + o(1)} \right)\log (1/(1 - p))\frac{n}{{\log n}}$$ ,     

On the chromatic number of random d-regular graphs

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 subgraph conditioning method to the number of equitable k-colourings, where a colouring is equitable if the number of vertices of each colour is equal.     

The chromatic number of dense random graphs

The chromatic number of a graph G is defined as the minimum number of colors required for a vertex coloring where no two adjacent vertices are colored the same. The chromatic number of the dense random graph where is constant has been intensively studied since the 1970s, and a landmark result by Bollobás in 1987 first established the asymptotic value of . Despite several improvements of this result, the exact value of remains open. In this paper, new upper and lower bounds for are established. These bounds are the first ones that match each other up to a term of size o(1) in the denominator: they narrow down the coloring rate of to an explicit interval of length o(1), answering a question of Kang and McDiarmid.     

Distinguishing chromatic number of random Cayley graphs

The distinguishing chromatic number of a graph $G$, denoted ${\chi }_D\left(G\right)$, is defined as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism of $G$ fixes each color class of $G$. In this paper, we consider random Cayley graphs $\Gamma$ defined over certain abelian groups $A$ with $\left|A\right|=n$, and show that with probability at least $1?n^{?\Omega (logn)}$, ${\chi }_D\left(\Gamma \right)\le \chi \left(\Gamma \right)+1$.     

The game chromatic number of random graphs

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 study, 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 same order of magnitude. We also study the game chromatic number of random bipartite graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The chromatic number of random Borsuk graphs

We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere S^d in , and a pair of vertices is connected if the Euclidean distance between them is at least 2??. We are interested in the chromatic number of this graph as n tends to infinity. It is not too hard to see that if ?>0 is small and fixed, then the chromatic number is d+2 with high probability. We show that this holds even if ?→0 slowly enough. We quantify the rate at which ? can tend to zero and still have the same chromatic number. The proof depends on combining topological methods (namely the Lyusternik–Schnirelman–Borsuk theorem) with geometric probability arguments. The rate we obtain is best possible, up to a constant factor—if ?→0 faster than this, we show that the graph is (d+1)‐colorable with high probability.25     

On embedding of finite distance graphs with large chromatic number in random graphs

The paper studies the problem indicated in the title, which arises in connection with the well-known Nelson–Erdös–Hadwiger problem of finding the chromatic number of the Euclidean space.     

The concentration of the chromatic number of random graphs

We prove that for every constant >0 the chromatic number of the random graphG(n, p) withp=n ^–1/2– is asymptotically almost surely concentrated in two consecutive values. This implies that for any <1/2 and any integer valued functionr(n)O(n ) there exists a functionp(n) such that the chromatic number ofG(n,p(n)) is preciselyr(n) asymptotically almost surely.Research supported in part by a USA Israeli BSF grant and by a grant from the Israel Science Foundation.Research supported in part by a Charles Clore Fellowship.     

On the chromatic number of circulant graphs

Given a set D of a cyclic group C, we study the chromatic number of the circulant graph G(C,D) whose vertex set is C, and there is an edge ij whenever i−j∈D∪−D. For a fixed set D={a,b,c:a<b<c} of positive integers, we compute the chromatic number of circulant graphs G(Z_N,D) for all N≥4bc. We also show that, if there is a total order of D such that the greatest common divisors of the initial segments form a decreasing sequence, then the chromatic number of G(Z,D) is at most 4. In particular, the chromatic number of a circulant graph on Z_N with respect to a minimum generating set D is at most 4. The results are based on the study of the so-called regular chromatic number, an easier parameter to compute. The paper also surveys known results on the chromatic number of circulant graphs.     

On the chromatic number of q-Kneser graphs

We show that the q-Kneser graph qK _2k:k (the graph on the k-subspaces of a 2k-space over GF(q), where two k-spaces are adjacent when they intersect trivially), has chromatic number q ^k ?+?q ^k?1 for k?=?3 and for k < q log q ? q. We obtain detailed results on maximal cocliques for k = 3.     

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

A crucial step in the Erdös-Rényi (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 Erdös and Rényi (1960). As a consequence, it is proved that the chromatic number is 3 for almost all graphs whenp=1/n.Research supported U.S. National Science Foundation Grants DMS-8303238 and DMS-8403646. The research was conducted on an exchange visit by Professor Wierman to Poland supported by the national Academy of Sciences of the USA and the Polish Academy of Sciences.     

On the locating chromatic number of Kneser graphs

Let c be a proper k-coloring of a connected graph G and Π=(C₁,C₂,…,C_k) be an ordered partition of V(G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c_Π(v):=(d(v,C₁),d(v,C₂),…,d(v,C_k)), where d(v,C_i)=min{d(v,x)|x∈C_i},1≤i≤k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ_L(G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ_L(KG(n,2))=n−1 for all n≥5. Then, we prove that χ_L(KG(n,k))≤n−1, when n≥k². Moreover, we present some bounds for the locating chromatic number of odd graphs.