A new coloring theorem of Kneser graphs

In 1997, Johnson, Holroyd and Stahl conjectured that the circular chromatic number of the Kneser graphs KG(n,k) is equal to the chromatic number of these graphs. This was proved by Simonyi and Tardos (2006) [13] and independently by Meunier (2005) [10], if χ(KG(n,k)) is even. In this paper, we propose an alternative version of Kneser's coloring theorem to confirm the Johnson-Holroyd-Stahl conjecture.     

Local coloring of Kneser graphs

A local coloring of a graph G is a function c:V(G)→N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥m_S, where m_S is the number of edges of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ_?(c). The local chromatic number of G is χ_?(G)=min{χ_?(c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K(n,k) for some values of n and k is determined.     

On the b-chromatic number of Kneser graphs

In this note, we prove that for any integer n≥3 the b-chromatic number of the Kneser graph KG(m,n) is greater than or equal to . This gives an affirmative answer to a conjecture of [6].     

Independence number of products of Kneser graphs

We study the independence number of a product of Kneser graph $K(n,k)$ with itself, where we consider all four standard graph products. The cases of the direct, the lexicographic and the strong product of Kneser graphs are not difficult (the formula for $\alpha (K(n,k)⊠K(n,k))$ is presented in this paper), while the case of the Cartesian product of Kneser graphs is much more involved. We establish a lower bound and an upper bound for the independence number of $K(n,2)\square K(n,2)$, which are asymptotically tending to $n^3∕3$ and $3n^3∕8$, respectively. The former is obtained by a construction, which differs from the standard diagonalization procedure, while for the upper bound the $\ell$-independence number of Kneser graphs can be applied. We also establish some constructions in odd graphs $K(2k+1,k)$, which give a lower bound for the 2-independence number of these graphs, and prove that two such constructions give the same lower bound as a previously known one. Finally, we consider the $s$-stable Kneser graphs $K{(ks+1,k)}_{s-stab}$, derive a formula for their $\ell$-independence number, and give the exact value of the independence number of the Cartesian square of $K{(ks+1,k)}_{s-stab}$.     

On the diameter of generalized Kneser graphs

Let r, k be positive integers, s(<r), a nonnegative integer, and n=2r-s+k. The set of r-subsets of [n]={1,2,…,n} is denoted by [n]^r. The generalized Kneser graph K(n,r,s) is the graph whose vertex-set is [n]^r where two r-subsets A and B are joined by an edge if |A∩B|?s. This note determines the diameter of generalized Kneser graphs. More precisely, the diameter of K(n,r,s) is equal to , which generalizes a result of Valencia-Pabon and Vera [On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].     

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.     

Distance graphs with maximum chromatic number

The distance graph G(D) has the set of integers as vertices and two vertices are adjacent in G(D) if their difference is contained in the set D⊂Z. A conjecture of Zhu states that if the chromatic number of G(D) achieves its maximum value |D|+1 then the graph has a triangle. The conjecture is proven to be true if |D|?3. We prove that the chromatic number of a distance graph with D={a,b,c,d} is five only if either D={1,2,3,4k} or D={a,b,a+b,b-a}. This confirms a stronger version of Zhu's conjecture for |D|=4, namely, if the chromatic number achieves its maximum value then the graph contains K₄.     

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).     

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.     

The chromatic number of infinite graphs — A survey

We survey some old and new results on the chromatic number of infinite graphs.     

Multichromatic numbers,star chromatic numbers and Kneser graphs

We investigate the relation between the multichromatic number (discussed by Stahl and by Hilton, Rado and Scott) and the star chromatic number (introduced by Vince) of a graph. Denoting these by χ* and η*, the work of the above authors shows that χ*(G) = η*(G) if G is bipartite, an odd cycle or a complete graph. We show that χ*(G) ≤ η*(G) for any finite simple graph G. We consider the Kneser graphs , for which χ* = m/n and η*(G)/χ*(G) is unbounded above. We investigate particular classes of these graphs and show that η* = 3 and η* = 4; (n ≥ 1), and η* = m - 2; (m ≥ 4). © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 137–145, 1997     

Random regular graphs of non-constant degree: Concentration of the chromatic number

In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model G_n,d 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 G(n,p) with . Our proof is largely based on ideas of Alon and Krivelevich who proved this two-point concentration result for G(n,p) for p=n^−δ where δ>1/2. The main tool used to derive such a result is a careful analysis of the distribution of edges in G_n,d, relying both on the switching technique and on bounding the probability of exponentially small events in the configuration model.     

5-chromatic strongly regular graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic. In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.     

A new approach to the chromatic number of the square of Kneser graph K(2k+1,k)

The vertices of Kneser graph $K(n,k)$ are the subsets of $\{1,2,\dots ,n\}$ of cardinality $k$, two vertices are adjacent if and only if they are disjoint. The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ with two vertices adjacent if their distance in $G$ is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when $n=2k+1$. It is believed that $\chi \left(K^2(2k+1,k)\right)=2k+c$ where $c$ is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: $8k∕3+20∕3$ for 1$k\ge 3$ (Kim and Park, 2014) and $32k∕15+32$ for $k\ge 7$ (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to $\chi \left(K^2(2k+1,k)\right)\le 5k∕2+c$, where $c$ is a constant in $\{5∕2,9∕2,5,6\}$, depending on $k\ge 2$.