Chromatic Number of Random Kneser Hypergraphs

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}(\rho) and proved that, in many cases, the chromatic numbers of the random Kneser graph \KG_{n,k}(\rho) 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}(\rho) as a very interesting problem. With the help of $\Z_p$-Tucker lemma, a combinatorial generalization of the Borsuk-Ulam theorem, we generalize Kupavskii's result to random general Kneser hypergraphs by introducing an almost surely lower bound for the chromatic number of them. Roughly speaking, as a special case of our result, we show that the chromatic numbers of the random Kneser hypergraph \KG^r_{n,k}(\rho) and the Kneser hypergraph \KG^r_{n,k} are almost surely closed in many cases. Moreover, restricting to the Kneser and {S}chrijver graphs, we present a purely combinatorial proof for an improvement of Kupavskii's results. Also, for any hypergraph \HH, we present a lower bound for the minimum number of colors required in a coloring of \KG^r(\mathcal{H}) with no monochromatic $K_{t,\ldots,t}^r$ subhypergraph, where $K_{t,\ldots,t}^r$ is the complete $r$-uniform $r$-partite hypergraph with $t r$ vertices such that each of its parts has $t$ vertices. This result generalizes the lower bound for the chromatic number of \KG^r(\mathcal{H}) found by the present authors~[{\it On the chromatic number of general {K}neser hypergraphs. J. Combin. Theory, Ser. B, {\rm 2015}.}]

On random subgraphs of Kneser and Schrijver graphs

A Kneser graph $KG_{n,k}$ is a graph whose vertices are in one-to-one correspondence with $k$-element subsets of $[n],$ with two vertices connected if and only if the corresponding sets do not intersect. A famous result due to Lov\'asz states that the chromatic number of a Kneser graph $KG_{n,k}$ is equal to $n-2k+2$. In this paper we study the chromatic number of a random subgraph of a Kneser graph $KG_{n,k}$ as $n$ grows. A random subgraph $KG_{n,k}(p)$ is obtained by including each edge of $KG_{n,k}$ with probability $p$. For a wide range of parameters $k = k(n), p = p(n)$ we show that $\chi(KG_{n,k}(p))$ is very close to $\chi(KG_{n,k}),$ a.a.s. differing by at most 4 in many cases. Moreover, we obtain the same bounds on the chromatic numbers for the so-called Schrijver graphs, which are known to be vertex-critical induced subgraphs of Kneser graphs

Coloring general Kneser graphs and hypergraphs via high-discrepancy hypergraphs

We suggest a new method on coloring generalized Kneser graphs based on hypergraphs with high discrepancy and small number of edges. The main result is providing a proper coloring of K(n, n/2-t, s) in (4 + o(1))(s + t)^2 colors, which is produced by Hadamard matrices. Also, we show that for colorings by independent set of a natural type, this result is the best possible up to a multiplicative constant. Our method extends to Kneser hypergraphs as well.Comment: 9 page

Dold's Theorem from Viewpoint of Strong Compatibility Graphs

Let $G$ be a non-trivial finite group. The well-known Dold's theorem states that: There is no continuous $G$-equivariant map from an $n$-connected simplicial $G$-complex to a free simplicial $G$-complex of dimension at most $n$. In this paper, we give a new generalization of Dold's theorem, by replacing "dimension at most $n$" with a sharper combinatorial parameter. Indeed, this parameter is the chromatic number of a new family of graphs, called strong compatibility graphs, associated to the target space. Moreover, in a series of examples, we will see that one can hope to infer much more information from this generalization than ordinary Dold's theorem. In particular, we show that this new parameter is significantly better than the dimension of target space "for almost all free $\mathbb{Z}_2$-simplicial complex." In addition, some other applications of strong compatibility graphs will be presented as well. In particular, a new way for constructing triangle-free graphs with high chromatic numbers from an n-sphere $\mathbb{S}^n$, and some new results on the limitations of topological methods for determining the chromatic number of graphs will be given.Comment: arXiv admin note: text overlap with arXiv:1709.0645

Local Clique Covering of Graphs

A k-clique covering of a simple graph G, is an edge covering of G by its cliques such that each vertex is contained in at most k cliques. The smallest k for which G admits a k-clique covering is called local clique cover number of G and is denoted by $lcc(G)$. Local clique cover number can be viewed as the local counterpart of the clique cover number which is equal to the minimum total number of cliques covering all edges. In this paper, several aspects of the problem are studied and its relationships to other well-known problems are discussed. Moreover, the local clique cover number of claw-free graphs and its subclasses are notably investigated. In particular, it is proved that local clique cover number of every claw-free graph is at most $c\Delta / \log\Delta$, where $\Delta$ is the maximum degree of the graph and $c$ is a universal constant. It is also shown that the bound is tight, up to a constant factor. Furthermore, it is established that local clique number of the linear interval graphs is bounded by $\log\Delta + 1/2 \log \log\Delta + O(1)$. Finally, as a by-product, a new Bollobas-type inequality is obtained for the intersecting pairs of set systems

Greedy online colouring with buffering

We consider the problem of online graph colouring. Whenever a node is requested, a colour must be assigned to the node, and this colour must be different from the colours of any of its neighbours. According to the greedy algorithm the node is coloured by the colour with the smallest possible $k$. The goal is to use as few colours as possible. We propose an algorithm, where the node is coloured not immediately, but only after the collection of next requests stored in the buffer of size $j$. In other words, the first node in the buffer is coloured definitively taking into account all possible colourisations of the remaining nodes in the buffer. If there are $r$ possible corrected colourings, then the one with the probability $1/r$ is chosen. The first coloured node is removed from the buffer to enable the entrance of the next request. A number of colours in a two examples of graphs: crown graphs and Kneser graphs have been analysed

Hypergraphs with many Kneser colorings (Extended Version)

For fixed positive integers $r, k$ and $\ell$ with $1 \leq \ell < r$ and an $r$-uniform hypergraph $H$, let $\kappa (H, k,\ell)$ denote the number of $k$-colorings of the set of hyperedges of $H$ for which any two hyperedges in the same color class intersect in at least $\ell$ elements. Consider the function \KC(n,r,k,\ell)=\max_{H\in{\mathcal H}_{n}} \kappa (H, k,\ell) , where the maximum runs over the family ${\mathcal H}_n$ of all $r$-uniform hypergraphs on $n$ vertices. In this paper, we determine the asymptotic behavior of the function \KC(n,r,k,\ell) for every fixed $r$, $k$ and $\ell$ and describe the extremal hypergraphs. This variant of a problem of Erd\H{o}s and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erd\H{o}s--Ko--Rado Theorem on intersecting systems of sets [Intersection Theorems for Systems of Finite Sets, Quarterly Journal of Mathematics, Oxford Series, Series 2, {\bf 12} (1961), 313--320].Comment: 39 page

Sharp bounds for the chromatic number of random Kneser graphs

Given positive integers $n\ge 2k$, the Kneser graph $KG_{n,k}$ is a graph whose vertex set is the collection of all $k$-element subsets of the set $\{1,\ldots, n\}$, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lov\'asz, states that the chromatic number of $KG_{n,k}$ is equal to $n-2k+2$. In this paper, we study the chromatic number of the {\it random Kneser graph} $KG_{n,k}(p)$, that is, the graph obtained from $KG_{n,k}$ by including each of the edges of $KG_{n,k}$ independently and with probability $p$. We prove that, for any fixed $k\ge 3$, $\chi(KG_{n,k}(1/2)) = n-\Theta(\sqrt[2k-2]{\log_2 n})$, as well as $\chi(KG_{n,2}(1/2)) = n-\Theta(\sqrt[2]{\log_2 n \cdot \log_2\log_2 n})$. We also prove that, for any fixed $l\ge 6$ and $k\ge C\sqrt[2l-3]{\log n}$, we have $\chi(KG_{n,k}(1/2))\ge n-2k+2-2l$, where $C=C(l)$ is an absolute constant. This significantly improves previous results on the subject, obtained by Kupavskii and by Alishahi and Hajiabolhassan. We also discuss an interesting connection to an extremal problem on embeddability of complexes

On the stability of the Erd\H{o}s-Ko-Rado theorem

Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp threshold result for this question in certain regimes. Since an independent set in the Kneser graph is the same as a uniform intersecting family, this gives us a random analogue of the Erd\H{o}s-Ko-Rado theorem.Comment: 17 pages, fixed misprints, Journal of Combinatorial Theory, Series

On Dynamic Coloring of Graphs

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant $c$ such that for every $k$-regular graph $G$, $\chi_d(G)\leq \chi(G)+ c\ln k +1$. Also, we introduce an upper bound for the dynamic list chromatic number of regular graphs